UC-NRLF 


B   M   550   70E 


Euclid's    Parallel  Postulate: 

Its  Nature,  Validity,  and  Place 

In  Geometrical  Systems. 


THESIS  PRESENTED  TO  THE   PHILOSOPHICAL  FACULTY   OF 

YALE  UNIVERSITV  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY. 


JOHN  WILLIAM  WITHERS,  Ph.D. 

Principal  of  the  Yeatman  High  School,  St.  Louis,  Mo. 


CHICAGO 
THE  OPEN  COURT  PUBLISHING  COMPANY 

LONDON  AGENTS 

Kegan  Paul,  Trench,  Trubner  &  Co.,  Ltd. 

1905. 


^^ 


Copyright  1905 


THE  OPEN  COURT  PUBLISHING  CO. 
Chicago 


PREFACE. 

The  parallel  postulate  is  the  only  distinctive  char- 
acteristic of  Euclid.  To  pronounce  upon  its  valid- 
ity and  general  philosophical  significance  without 
endeavoring  to  know  what  Non-Euclideans  have 
done  would  be  an  inexcusable  blunder.  For  this 
reason  I  have  given  in  the  following  pages  what 
might  otherwise  seem  to  be  an  undue  prominence 
to  the  historical  aspect  of  my  general  problem. 

In  the  last  chapter,  the  positions  taken  are  only 
briefly  defended,  because  they  seem  to  flow  directly 
and  naturally  from  results  previously  won. 

I  have  included  in  the  bibliography  such  works 
as  are  mentioned  in  the  body  of  the  thesis,  and 
have  not  aimed  at  making  a  complete  list.  More 
complete  biographies  of  Hyperspace  and  non- 
Euclidean  Geometry  are  those  of  Halsted  and  Bon- 
ola,  which  I  have  mentioned  in  my  list. 

My  obligations  not  elsewhere  explicitly  acknowl- 
edged are  chiefly  to  Professor  Geo.  T.  Ladd,  at 
whose  suggestion  this  study  was  undertaken,  and 
under  whose  sympathetic  direction  it  has  attained 
its  present  form.  I  am  also  indebted  to  Dr.  E.  B. 
Wilson  for  light  upon  certain  mathematical  aspects 
of  the  problem. 

Neiv  Haven,  Connecticut,  April,  1904. 


CONTENTS. 


CHAPTER  PAGE 

I.  The  Pre-Lobatchewskian  Struggle  With 
THE  Parallel  Postulate. 
Historical  origin  of  the  Postulate.  Attempts 
to  dispense  with  it.  Substitution  of  different 
definitions.  Substitution  of  different  postu- 
lates. Efforts  to  prove  the  Parallel  Postulate 
on  a  basis  of  Euclid's  other  assumptions   .      .       1-20 

II.  The  Discovery  and  Development  of  Non- 
Euclidean  Systems. 
The  clue.  Invention  of  consistent  geome- 
tries independent  of  the  Parallel  Postulate. 
The  problem  generafized.  Attention  directed 
through  projective  methods  to  the  qualitative 
aspects  "of  spaCfe.     Recent  developments  23-60 

III.  General  Orientation  of  the  Problem. 
General  significance  of  the  historical  develop- 
ment of  non-Euclidean  geometry.  The  motive. 
Complex  character  of  the  problem  and  what 
its  solution  requires.  Delimitation  of  the  pres- 
ent study 63-76 

IV.  Psychology  of  the  Parallel  Postulate  and 

Its  Kindred  Conceptions. 
General  observations  as  to  the  development  of 
our  space  conception.  Empirical  sources  of 
special  geometrical  conceptiohs.  Of  measure- 
ment and  the  consciousness  of  Homogeneity 
and  Free  Mobility.  Of  the  straight  TTne"  Of 
Surfaces.  Of  the  Parallel  Postulate ;  its  em- 
pirical  sources ;    its  direct   relations  to  sense- 

V 


vi  CONTENTS 

CHAPTER  PAGE 

perception;  its  complex  nature;  its  relations 
to  the  conception  of  the  straight  line ;  empirical 
nature  of  the  straight  line.  The  validity  of 
the  Parallel  Postulate  not  to  be  settled  in  ab- 
solute fashion  by  any  empirical  investigation     79-123 

V.  The  Nature  and  Validity  of  the  Parallel 
Postulate. 
The  number  and  variety  of  possible  geometries. 
Space  parameter  not  necessarily  constant  to 
accord  with  the  space  of  experience.  Distinc- 
tion between  the  facts  of  experience  and  the 
intellectual  constructs  formed  on  a  basis  of 
this  experience.  Conditions  imposed  upon 
these  constructs.  Non-Euclidean  systems  log- 
ically possible.  The  Parallel  Postulate  not  a 
priori  necessary.  How  judge  the  validity  of 
Euclid.  Difficulties  with  Poincare's  doctrine 
of  convenience.  Why  is  Euclid  most  con- 
venient ?  Kant's  Argument.  Its  failure  to  es- 
tablish the  necessary  validity  of  Euclid.  Must 
appeal  to  experience.  Methods  of  scientific 
testing  and  the  difficulties  which  attend  them. 
The  experimental  method  suggested  in  Ladd's 
A   Theory  of  Reality 127-152 

VI.  Resulting  Implications  as  to  the  Nature 
OF  Space. 
Distinctions  drawn  between  conceptions  of 
space  and  of  figures  in  space.  The  importance 
of  these  distinctions  pointed  out.  Varieties  of 
two-dimensional  spaces  and  of  tri-dimensional 
spaces  possible.  Possibility  of  a  fourth  di- 
mension.*^ Euclidean  and  Non-Euclidean  tri- 
dimensional spaces  independent  of  a  fourth 
dimen'sion.  What  this  implies.  The  space- 
constant  and  what  it  means.  The  impossibility 
of  comparing  different  space-constants  quan- 
titatively in  a  strict  geometrical  sense.     Log- 


CONTENTS 


ical  difficulties  of  the  conception  of  variable 
curvature  quantitatively  considered.  Diffi- 
culties due  to  abstract  conception.  Geometry 
must  fit  experience ;  not  experience  geometry. 
Homogeneity  does  not  mean  mere  logical  any- 
ness.  Distinction  between  geometrical  space 
conceptions  and  the  space  category.  The  a 
priori  element.     Summary   and   conclusion     .   155-173 

VII.    Bibliography 177-192 


PRE-LOBATCHEWSKIAN 
STRUGGLE 


WITH  THE 


PARALLEL    POSTULATE. 


CHAPTER  I. 

THE    PRE-LOBATCHEWSKIAN    STRUGGLE. 

I.  Historical  Origin  of  the  Parallel  Postulate. — 
Before  the  time  of  Euclid  the  science  of  geometry- 
was  already  well  advanced  in  Greece.  Pythagoras 
and  his  immediate  followers,  by  perfecting  a  theory 
of  ratio  and  proportion,  and  by  the  study  of  areas 
and  the  introduction  of  irrational  quantities,  had 
brought  the  subject  so  prominently  before  the  Greek 
mind  that  no  subsequent  philosopher  could  afford 
to  neglect  it.  Accordingly,  Zeno  and  Democritus, 
Anaxagoras  and  Hippias,  Plato  and  Aristotle,  and 
the  great  body  of  thinkers  who  were  disciples  of 
these  more  or  less  extensively  devoted  themselves 
to  its  study. 

The  duplication  of  the  cube,  the  quadrature  of 
the  circle,  the  trisection  of  the  angle,  were  all  vigor- 
ously attacked  and  out  of  the  struggle  certain  new 
and  important  conceptions  arose.  Certain  lines  of 
plane  and  double  curvature  were  invented,  impor- 
tant properties  of  conies  were  discovered,  and  the 
notion  of  infinity  introduced.  Methods  of  research 
and    geometrical    exposition    were   also    accurately 


2         THE  PRE-LOBATCHEIVSKIAN  STRUGGLE 

Studied,  among  them  the  method  of  reduction  of 
Hippocrates,  the  analytical  method  of  Plato,  and 
the  method  of  exhaustions  of  Eudoxus.  i\dded  to 
these  the  diorism  of  Leon,  the  determination  by 
Menjechmus  of  the  necessary  conditions  for  the 
invertibility  of  a  theorem  which  affords  a  fruitful 
method  of  enlarging  the  number  of  propositions, 
the  introduction  of  formal  logic  and  the  powerful 
influence  of  the  dialectics  of  Socrates  and  the 
Sophists,  all  contributed  to  make  possible  that  re- 
markable outburst  of  mathematical  genius  which 
has  been  fittingly  styled  "  The  Golden  Age  of  Greek 
Geometry/' 

It  is  obvious  then  in  view  of  this  remarkable  de- 
velopment that  Euclid  was  by  no  means  the  author 
of  all  the  demonstrations  contained  in  his  Elements. 
It  is  impossible  to  state  exactly  what  he  did  con- 
tribute. In  the  whole  collection  there  is  only  one 
proofs  (I,  47)  which  is  directly  ascribed  to  him. 
Of  a  few  things,  however,  we  are  reasonably  sure. 
Euclid  brought  to  irrefutable  demonstration  propo- 
sitions which  had  been  previously  less  rigorously 
proved.^      The   selection   and    arrangement   of   the 

^  Gow's  History  of  Greek  Geometry,  Cambridge,  1884,  p. 
198. 

-  Proclus,  at  close  of  the  Eudemian  Summary.  Eudemas, 
a  pupil  of  Aristotle,  wrote  a  history  of  Geometry  which  has 
been  lost,  but  Proclus,  in  his  commentaries  on  Euclid,  gives 
an  abstract  or  summary  of  it.  and  this  is  the  most  trustworthy 
information  we  have  regarding  early  Greek  Geometry. 


THE  FRE-LOBATCHEllSKJAA'  STRUGGLE  3 

propositions  is  his.''  He  chose  the  theorems  and 
demonstrations  which  should  form  a  part  of  his 
system.  Many  available  demonstrations  were  cer- 
tainly rejected."*  We  may  attribute  to  his  delib- 
erate choice  the  distinctive  characteristics  of  the 
book  as  a  whole.  We  owe  to  him  that  orderly 
method  of  proof  which  proceeds  by  statement,  con- 
struction, proof,  conclusion,  even  to  the  final  Q.  E. 
D.  (oTTcp  18a  Set^oi)  of  the  modern  text.  He  is 
responsible  too  for  that  peculiar  logical  design 
of  the  book  which  proceeds  always  from  a  few 
definitions,  postulates  and  common  notions,  or 
axioms,  by  sure  steps  which  are  always  of  precisely 
the  same  kind  until  every  link  in  the  argument  from 
premises  to  conclusion  is  securely  forged. 

Euclid  set  at  the  beginning  of  his  text  certain 
definitions,  postulates  and  common  notions  ^  which 
should  serv^e  as  the  foundation  for  his  system. 
Was  the  parallel  postulate  one  of  this  number? 

There  is  but  one  way  to  answer  this  question, 
and  that  is  to  go  back  to  the  earliest  editions  of 
Euclid  at  present  accessible  and  observe  whether 
they  contain  it  or  not.  In  all  these  earlier  editions 
there  is  practical  agreement  in  regard  to  the  defini- 


'  Proclus,  Friedlein's  Edition,  p.  69. 
*  Compare  Gow,  op.  cit.  pp.  iq8  fF. 

'^  Euclid  did  not  use  the  term  axiom.     This  was  introduced 
by  Proclus. 


4         THE  PRE-LOBATCHEIVSKIAN  STRUGGLE 

tions  of  Euclid.  As  far  back  as  lOO  b.  c.  Heron's^ 
"  Definitions  "  appear  in  the  same  number  and  in 
essentially  the  same  form,  though  not  in  the  same 
order  as  we  have  them  now.  Among  the  postulates 
and  Common  Notions,'^  however,  there  is  consider- 
able fluctuation.  Many  editions  give  three  postu- 
lates and  twelve  axioms.  The  first  nine  axioms 
relate  to  all  kinds  of  magnitudes;  these  remain  in 
all  editions  essentially  unchanged ;  but  the  last  three 
which  relate  to  space  only  and  are  thus  distinctively 
geometrical  fluctuate  in  a  most  interesting  way. 
They  are  as  follows :  lo,  Two  straight  lines  cannot 
inclose  a  space;  ii.  All  right  angles  are  equal;  12, 
If  a  straight  line  meet  two  straight  lines,  so  as  to 
make  the  two  interior  angles  on  the  same  side  of 
it  taken  together  less  than  two  right  angles,  these 
straight  lines  being  continually  produced,  shall  at 
length  meet  on  that  side  on  which  are  the  angles 
which  are  less  than  two  right  angles.^  In  nearly  all 
modern  editions,  except  the  most  recent,  these  are 
called  "  axioms  "  and  classed,  as  here  indicated,  in 
the  same  category  with  the  other  nine.  In  the  older 
manuscripts,  however,  no  such  blunder  is  commit- 
ted.    It  seems  hardly  possible  to  account  for  this 

^  Simon's  Euclid  Sec.  6.     Die  Kommentatoren  des  Euclid. 

''  Euclid's  expression  is  Kotvai  ivvowLL- 

*  In  Clavius  this  is  the  13th  axiom.  Robert  Simson,  on 
whose  edition  modern  English  texts  are  usuallj'  based,  calls 
it  the  I2th  axiom;  Bolyai  and  many  others  call  it  the  nth 
axiom. 


THE  PRE-LOBATCHEWSKIAN  STRUGGLE         5 

unless  we  assume  that  Euclid  himself  drew  a  dis- 
tinction between  them.     Of  these  older  manuscripts 
by    far    the    greater    number    place    these    axioms 
among  the   postulates    where   they   rightly   belong. 
The  Vatican  Manuscript  ^   gives   "axiom"    10  as 
postulate  6.     Proclus  omits  it  altogether  with  the 
significant  remark  that  it  is  really  a  theorem  which 
ought  to  be  proved,  and  actually  attempts  to  prove 
it   in   book   I,    proposition   4.      He   also   gives   the 
parallel  "  axiom  "  as  postulate  5,  but  claims  that  it, 
too,  should  be  proved,   and  quotes  Germinus  ^^  in 
support   of  this  view.     Thus  it   appears   not  only 
that  this  famous  postulate  is  at  least  of  classical 
origin,  but  also  that  the  Greeks  themselves  better 
understood   its   true  nature   than   the   majority   of 
modern  writers  have  done. 

But  we  have  not  yet  answered  our  question;  did 
Euclid  himself  actually  make  use  of  this  postulate? 
If  so,  v/as  it  his  own  invention  or  did  he  borrow  it 
from  another?  The  latter  question  we  can  not 
definitely  answer  since  we  have  no  means  of  knowing 
how  many  of  the  propositions  which  involve  this 
assumption  were  first  demonstrated  by  Euclid  hi 


imi- 


9  Discovered  by  Napoleon  in  Rome  in  the  early  part  of  last 
century  and  brought  to  France.  It  was  edited  in  Paris  by 
R  Peyrard  (1814-1818).  who  thought  that  he  had  here  an 
edition  of  Euclid  more  ancient  than  Theon's  (about  380  A  D  ) 
upon  which  many  of  the  early  MSS.  which  first  came  to  light 
"laimed  to  be  based. 

'"  Germinus  wrote  about  60  B.  C. 


6         THE  PRE-LOBATCHEWSKIAN  STRUGGLE 

self;  nothing  like  this  postulate  is  mentioned  how- 
ever by  any  previous  writer. 

Regarding  the  first  question  we  can  speak  with 
more- confidence.  The  researches  of  Peyrard  show 
that  in  the  earliest  manuscripts  now  accessible,  this 
postulate  does  not  appear  among  the  other  postulates 
and  common  notions  at  the  beginning  of  the  text, 
but  is  found  in  the  demonstration  of  Proposition  29 
where  it  is  introduced  to  support  the  proof  of  the 
equality  of  the  alternate  angles  of  parallel  lines. 
Euclid  himself  then  not  only  employed  this  postulate 
but  the  position  in  which  he  placed  it  seems  to  indi- 
cate clearly  that  he  appreciated  the  difficulties  which 
its  use  involves.  Surely  one  who  had  formulated  a 
system  so  rigorous,  who  was  master  of  a  logic  so 
keen  and  true  that  the  most  critical  efforts  of  modern  | 
thought  have  not  destroyed  but,  on  the  contrary, 
have  only  strengthened  his  claims  to  rigor,  could  not 
have  passed  over  such  a  manifest  begging  of  the 
question  as  appears  upon,  the  very  face  of  this  postu- 
late as  he  himself  phrased  it,  without  having  first 
made  a  desperate  effort  to  prove  it.  Euclid  makes 
no  attempt  as  did  later  waiters  to  conceal  the  diffi- 
culty under  the  cloak  of  a  subtle  phraseology.  He 
states  it  frankly  as  a  petitio  principii  of  the  baldest 
type.  It  must  then  have  appeared  to  him  not  as  an 
"  axiomatic  "  truth,  but  as  a  theorem  calling  for 
demonstration.  Euclid  proves  propositions  more 
obvious  bv   far.      He  even   demonstrates  that  two 


THE  PRE-LOBArCIlEWSKIAN  STRUGGLE  7 

sides  of  a  triangle  are  greater  than  the  third  side, 
a  proposition  which  the  Epicureans  derided  as  being 
"  manifest  even  to  asses."  '^ 

The  position  of  the  postulate  seems  to  indicate 
that  EucHd  struggled  on  as  far  as  possible  without 
it  and  postulated  it  finally  only  because  he  could 
neither  prove  it  nor'  proceed  any  further  without  it. 
Moreover,  the  astronomical  system  of  Eudoxus  and 
the  writings  of  x\ntolycus  make  it  also  apparent 
that  Euclid  must  have  had  some  knowledge  of  sur- 
face spherics  and  was  therefore  familiar  with  tri- 
angles whose  angle  sum  contradicts  the  truth  of 
this  postulate. 

II.  Attempts  to  Dispense  unth  the  Postulate. — 
The  intersection  of  two  slowly  converging  straight 
lines  lies  of  course  beyond  the  province  of  observa- 
tion or  construction.  Hence  it  is  obvious  why  the 
successors  of  Euclid,  habituated  by  him  to  strict 
logical  rigor,  should  have  found  fault  with  the 
i:)arallel  postulate  and  put  forth  their  utmost  en- 
deavors to  dispose  of  it  in  one  way  or  another.  In 
1621  Sir  Henry  Saville^^  wrote:  "In  pnlcherrinw 
Geouietricr  eorpore  duo  sunt  iiarvi,  duce  lobes  nee 
quod  seiani  plures  in  quibus  elucendis  et  eniacu- 
l  end  is  euni  veferum  turn  reeentioruni  vigilavit  in- 
dustria."     One  of  these  "blemishes"  was  the  par- 

11  Proclus.    op.    cit..    also    Cajori,    History    of    Elementary 
Mathematics,  p.  74. 

^Lectures  on  Euclid,  published  at  Oxford,  1621. 


8         THE  PRE-LOBATCHEWSKIAN  STRUGGLE 

allel  postulate,  the  other  EucUd's  theory  of  propor- 
tion. Under  the  title  "  Parallel "  in  the  "  En- 
cyclopadie  der  Wisscnschaften  und  Kiinste,"  pub- 
lished at  Leipzig  in  i^sS.  Sohncke  says  that  "in 
Mathematics  there  is  nothing  over  which  so  much 
has  been  spoken,  written  and  striven,  and  all  so  far 
without  reaching  a  definite  result  and  decision." 
Appended  to  this  article  there  is  a  carefully  pre- 
pared list  of  ninety-two  authors  who  had  dealt  with 
the  problem.  These  quotations  show  the  extent  to 
which  these  earlier  efforts  were  carried.  Indeed  it 
appears  that  almost  every  writer  on  geometry  ofj 
any  note  from  Euclid  to  Sohncke  had  given  mor 
or  less  attention  to  this  difficult  subject. 

These  earlier  endeavors  struck  out  in  various 
directions  which  we  shall  now  briefly  state  and  con 
sider.  Some  attempted  to  avoid  the  difficulty 
through  a  new  definition  of  parallel  lines ;  by  others 
new  assumptions  which  were  considered  less  faulty 
were  substituted  for  Euclid's.  These  in  reality  only 
concealed  the  difficulty ;  they  did  not  remove  it.  A 
third  class  attempted '  to  deduce  the  theory  of 
parallels  from  Euclid's  other  postulates,  by  reason- 
ing upon  the  nature  of  the  straight  line  and  the 
plane  angle.  These  were  by  far  the  most  desperate 
attempts.  Finally  there  were  those  who  decided 
that  if  this  postulate  is  dependent  upon  the  other 
assumptions  which  constitute  the  foundations  of 
Euclid  we  shall  by  denying  it  and  maintaining  them, 


THE  PRE-LOBATCHEWSKIAN  STRUGGLE         9 

become  ultimately  involved  in  contradiction.  It 
was  this  method  of  procedure  which  resulted  in  the 
first  establishment  of  a  non-Euclidean  geometry. 
We  shall  consider  these  attempts  in  the  order 
named. 

(i)      The  Substitution  of  Different  Definitions. 

Euclid's  own  definition  was,  that  parallel  lines 
are  straight  lines  which  lie  in  the  same  plane  and 
will  not  meet  however  far  produced.  This  defini- 
tion is  perhaps  still  best  for  elementary  geometry. 
In  1525  Albrecht  Diirer/^  a  German  painter,  pro- 
posed the  familiar  definition  that  parallel  lines  are 
straight  lines  which  are  everywhere  equally  distant. 
Clavius^'*  substituted  for  this  the  assumption  that 
a  line  which  is  everywhere  equidistant  from  a  given 
straight  line  in  the  same  plane  is  itself  straight. 
A.nother  definition  which  is  often  preferred  because 
3f  its  apparent  simplicity  is.  that  parallel  lines  are 
traight  lines  which  have  the  same  direction.  This 
definition  possesses  the  peculiar  advantage  that 
;hose  who  adopt  it  have  no  further  difficulty;  for 
:hey  find  no  necessity  to  assume  the  parallel  postu- 
ate  or  anything  equivalent  to  it.  This  is  a  great 
idvantage,  certainly ;  but,  as  a  matter  of  fact,  any 
me  of  these  definitions,  though  apparently  more 
idvantageous  than  Euclid's,  is  in  reality  more  com- 
)lex  and  less  satisfactory.     The  first  two  make  use 

13  Cajori,  p.  266. 

1*  Edition  of  Euclid,  1574. 


10       THE  PKE-LOBATCHEWSKIAN  STRUGGLE 

of  the  conception  of  distance.  This  of  course  in 
volves  measurement,  which  in  turn  embraces  the  L 
whole  theory  of  incommensurable  quantities  with  I  ^ 
its  entire  outfit  of  necessary  presuppositions  and, 
attendant  difficulties.  What  is  more,  these  defini- 
tions only  hold  for  Euclidean  geometry;  they  are 
not  true  for  pseudo-spherical  space  where  parallel 
lines  are  still  possible  and  where  Euclid's  definition 
is  still  valid.  The  objection  to  the  third  definition 
is  its  use  of  the  term  "  direction,"  a  word  which 
because  of  its  apparent  simplicity,  but  real  obscurity 
and  vagueness,  is  exceedingly  misleading  and 
troublesome.  For  example,  the  straight  line  is 
often  defined  as  one  which  does  not  change  its 
direction  at  any  point,  and  yet  this  same  line  is  said 
to  have  opposite  directions.  Again,  the  angle  is! 
sometimes  defined  as  a  difference  of  direction.  Mo- 
tion in  the  circumference  of  a  circle  is  said  to  be 
in  a  clockwise  or  counter-clockwise  direction,  and 
in  this  sense  a  point  may  move  all  round  the  cir- 
cumference without  changing  its  direction,  and  yet 
we  speak  of  this  same  circumference  as  a  line  which 
changes  its  direction  at  every  point.  Killing  has 
shown  that  the  word  direction  can  only  be  defined 
when  the  theory  of  parallels  is  already  presup- 
posed.^'' 

Many  other  definitions  have  been  proposed,  but 

'•''  Einfuehning    in   die   Grundlagen    der   Geometric,    Pader- 
born,  1898. 


THE  FRE-LOBATCHEIVSKIAN  STRUGGLE        ii 

hey  throw  no  light  upon  the  problem,  and  with  one 
xception  they  may  be  omitted.  This  exception  is 
he  definition  proposed  by  Kepler  ^^  and  De- 
argues/'^  which  is  that  parallel  lines  are  straight 
nes  which  have  a  common  point  at  infinity ;  or  "  If 
"v  be  a  point  without  a  given  indefinite  right 
ne  CD,  the  shortest  line  that  can  be  drawn  from 
to  it  is  perpendicular,  and  the  longest  line  is 
arallel  to  CD."  ^'^  This  definition  is  important  for 
rojective  geometry. 

(2)  The  substitution  of  different  postulates 
as  been  frequently  made.  Of  these  Playfair's  ^® 
)rmulation  that  "  Two  straight  lines  which  cut 
ne  another  cannot  both  be  parallel  to  the  same 
:raight  line,"  is  perhaps  the  least  objection- 
Die.  Cayley  ^'^  considered  this  statement  to  be 
Kiomatic. 
Nasir  Eddin  (i  201 -1274),  a  gifted  Persian 
tronomer,  in  an  edition  of  Euclid  subsequently 
rinted  in  Arabic  and  brought  out  in  Rome  in  1594, 
akes  the  following  assumption :  "  If  AB  is  per- 
ndicular  to  CD  at  C,  and  if  another  straight  line 


^^  Kepler's  Paralipomena,  1604. 

^  Brotiillon  Proiect,  1639. 

8  Stone's  Neiv  Mathematical  Dictionary,  London,  1743. 

»  Playfair  credits  this  axiom  to  Ludlum.  See  Halsted's 
tide  in  Science,  N.  S.  Vol.  XIII.,  No.  325,  March  22,  1902, 
t.  462-465. 

His   Presidential   Address,   Collected   Math.   Papers,  Vol. 

I.,  pp.  429-459.  ^  ^---^-rn-^*^ 

■/     ^'   or  THE  \ 

I   UNIVER81TYJ 


12       THE  PRE-LOBATCHEVVSKIAN  STRUGGLE 

EUF  makes  the  angle  EDC  acute,  then  the  perpen- 
diculars to  AB  comprehended  between  AB  and  EF, 
and  drawn  on  the  side  of  CD  toward  E,  are  shorter 
and  shorter,  the  further  they  are  from  CD."  Or 
in  general,  two  straight  lines  which  cut  a  thuu 
straight  line,  the  one  at  right  angles,  the  other  at 
some  other  angle,  will  converge  on  the  side  where 
the  angle  is  acute  and  diverge  where  it  is  obtuse. 
Nothing  is  here  said  as  to  whether  the  two  lines 
will,  or  w^ill  not,  eventually  meet ;  the  assumption 
is  therefore  as  valid  for  pseudo-spherical  as  it  is 
for  Euclidean  space. 

The  work  of  Nasir  Eddin  was  taken  up  by  John 
Wallis  and  communicated  in  a  Latin  translation  to 
the  mathematicians  at  Oxford  ^^  in  1651;  and  on 
the  evening  of  July.  11,  1663,  Wallis  himself  deliv- 
ered a  lecture  at  Oxford  ^^  in  w^iich  he  recom- 
mended for  Euclid's  postulate  the  assumption  of 
the  existence  of  similar  figures  of  different  sizes,  or 
to  quote  his  own  statement,  "  To  any  triangle  an- 
other triangle  as  large  as  you  please  can  be  drawn 
which  is  similar  to  the  given  triangle."  This  is 
easily  shown  to  be  equivalent  to  the  Euclidean  pos- 
tulate. Such  figures  are  impossible  in  any  form  of 
non-Euclidean  space.  Saccheri  proved  that  Euclid- 
ean geometry  can  be  rigidly  developed  if  the  exist- 

2'  Wallis.  Ot^cra  II.,  669-673. 

"-  EiiRel  and  Staeckel,  "  Die  Theorie  der  ParallcUinien  von 
Euclid  his  auf  Gauss,  Leipzig  1895.  pp.  21-30. 


THE  PRE-LOBATCHEWSKIAN  STRUGGLE        13 

ence  of  one  such  triangle,  unequal  but  similar  to 
another,  may  be  granted.  Carnot  and  La  Place  and, 
more  recently.  J.  Delboeuf,^^  have  proposed  the 
adoption  of  Wallis's  postulate. 

In  1833  1-  Perronet  Thompson  of  Cambridge 
Dublished  a  book  '^*  in  which  he  brilliantly  demon- 

trates  the  insufficiency  of  twenty-one  different  at- 
tempts to  dispdse  of  the  Parallel  postulate,  and 
:loses  the  volume  with  a  demonstration  of  his  own 

n  which  he  claims  to  "  establish  the  theory  of 
parallel  lines  without  recourse  to  any  principle  not 
3Tounded  on  previous  demonstration."  ^"^  This  en- 
deavor, however,  belongs  to  the  third  class  of  at- 
tempted solutions  which  we  must  now  very  briefly 

onsider. 

(3)  The  first  recorded  attempt  to  prove  the 
parallel  postulate  on  the  basis  of  Euclid's  other 
assumptions  was  that  of  Ptolemy  in  his  treatise  on 
pure  geoiiicfry.-''  This  proof  assumes  of  course 
the  validity  of  postulate  six,^'^  which  does  not  hold 
in  elliptic  s])ace  and  also  involves  the  untenable 
assertion  that,  in  the  case  of  parallelism,  the  sum 
Df  the  interior  angles  on  one  side  of  the  transversal 
must  be  the  same  as  that  upon  the  other  side. 

23  Engel  and  .Stacckel  op.  cit.  p.  19. 
-*  "  Geometry  without  Axioms." 
-■'■•  Quoted  from  the  title. 

2fi  Gow.  p.  301.     For  this  we  are  indebted  to  Proclus. 
-'  Of  the  Vatican  MSS. :     Two  straight  lines  can  not  en- 
close a  space. 


14       THE  PRE-LOBATCHEWSKIAN  STRUGGLE 

>  One  of  the  most  scientific  attempts  of  this  class 
was  that  of  Girolamo  Saccheri  in  his  volume  en- 
titled. "  Eiiclidis  ab  onini  nccvo  vindicatus,  sive 
conatus  gcomctriciis  quo  stabiliuntur  prima  ipsa 
univcrscc  gcomatricE  principia."  ^^  This  work  only 
recently  came  to  light.  As  late  as  1893  Professor 
Klein,  himself  an  able  contributor  to  the  knowledge 
of  Hyperspace  and  non-Euclidean  geometry,  had 
not  even  heard  of  Saccheri.  In  1889  E.  Beltrami, 
at  the  suggestion  of  the  Italian  Jesuit,  P.  Manga- 
notti.  published""  a  note"^  in  which  he  showed 
that  Saccheri  had  practically  wrought  out  a  non- 
Euclidean  geometry  almost  a  century  before  Lo- 
batchewsky  and  Bolyai.  Apparently  the  only 
thing  which  prevented  Saccheri  from  perceiving 
the  significance  of  his  discovery  was  his  blinding 
desire  to  "  vindicate  Euclid  from  every  fault.''  His 
statement  of  the  problem  shows  clearly  that  he  was 
on  the  right  road  to  discovery.  If  the  parallel 
axiom '"   is  involved  in  the  remaining  assumptions 

-^  Published  at  Milan,  where  he  was  president  of  the  Col- 
Icgio  di  Brera,  shortly  before  his  death  in  1733.  This  work 
is  now  exceedingly  rare,  the  only  copy  on  the  Western  Conti- 
nent, perhaps,  is  that  of  Professor  Halsted,  who  has  trans- 
lated it  into  English.  It  has  also  been  translated  into  German 
and  forms  a  part  of  Engel  and  Staeckel's  History  of  Parallel 
Theory  previously  referred  to. 

'-".'J//I  dcUa  Realc  Accademia  dci  Lined. 

■■"'  Un  prcctirsorc  italiane  di  Legendri  c  di  Lobatchewski. 

•'"  Saccheri  calls  it  an  axiom.  He  studied  Clavius's  edition, 
in  which  it  appears  as  the  i.nh  axiom. 


THE  PRE-LOBATCHEWSKIAN  STRUGGLE        15 

of  Euclid,  then  it  will  be  possible  to  prove  without 
its  aid  that  in  any  quadrilateral  ABCD  having  right 
angles  at  A  and  B  and  the  side  XC  equal  to  the 
side  DB,  the  angles  C  and  D  are  also  right  angle? 
and  in  that  event  the  assumption  that  C  and  D  are 
either  obtuse  or  acute  will  lead  to  contradiction.  He 
proves  that  these  angles  cannot  be  obtuse,  for  in 
that  case  Euclid's  axiom  that  two  straight  lines 
cannot  enclose  a  space  is  contradicted ;  but  when  he 
endeavors  to  prove  that  they  cannot  be  acute  he 
fails  of  his  purpose  for  in  this  case  he  does  not 
meet  with  any  contradiction. 

In  regard  to  the  angles  C  and  D  he  distinguishes 
three  hypotheses  as  follows :  ( i )  hypothesis  anguli 
recti,  (2)  hypothesis  anguli  ohtiisi,  and  (3)  hy- 
pothesis anguli  aciiti.  He  then  proves  that  if  either 
hypothesis  is  true  in  a  single  case  it  is  always  true,'^ 
that  in  a  right  triangle  the  sum  of  the  oblique  angles 
is  equal  to.  greater  than,  or  less  than,  a  right  angle 
according  as  the  hypothesis  is  anguli  recti,  anguli 
obtusi,  or  anguli  acutif'''  He  ne;xt  shows  that  in 
the  first  two  hypotheses  a  perpendicular  and  an 
oblique  to  the  same  straight  line  will  meet  if  suffi- 
ciently produced,^'*  hence  in  these  two  cases  Euclid's 
postulate  is  not  contradicted.^^     He  then  proceeds 


3-'  Propositions  y.,  VI.,  and  VII. 
•■'•*  Propositions  VIII.  and  IX. 
3*  Propositions  XI.  and  XII. 
35  Proposition  XIII. 


i6       THE  PRE- LOB  AT  CHEW  SKI  AN  STRUGGLE 

to  prove  that  according  as  the  triangle's  angle  sum 
is  equal  to,  greater  than,  or  less  than,  two  right 
angles  we  have  hypothesis  angiili  recti,  obtusi,  or 
acuti;^'^  and  that  with  the  hypothesis  anguli  aciiti 
we  can  draw  a  perpendicular  and  an  oblique  to  the 
same  straight  line  which  will  nowhere  meet  each 
other.3^ 

It  is  unnecessary  to  follow  him  further.  We  now 
have  enough  to  show  that  Saccheri  understood  the 
close  connection  between  the  parallel  postulate  and 
right  angles.  In  his  eager  quest  for  contradictions 
in  pursuit  of  the  hypothesis  anguli  acuti  he  prac- 
tically attained  without  knowing  it,  those  far  reach- 
ing conclusions  which  were  disclosed  a  century 
later.  But  on  the  very  verge  of  discovery,  being 
blinded  by  an  intellectual  bias  toward  the  tradi- 
tional view,  he  rejects  this  hypothesis  upon  the  un- 
satisfactory ground  that  it  is  incompatible  with  the 
nature  of  the  straight  line;  "  for."  says  he,  "  it  per- 
mits tl>e  assumption  of  different  kinds  of  straight 
lines  which  meet  at  infinity  and  have  there  a  com- 
mon perpendicular." 

Conclusions  very  like  the  foregoing  w^re  also 
reached  by  John  Henry  Lambert,  whom  Kant  calls 
"  dcr  nnverglcichliche  Mann."  Lambert  is  free 
from  that  strained  reverence  for  Euclid  which  char- 
acterized Saccheri.  and  consequently  advances  be- 

3"  Propositions  XV.  and  XVI. 
"  Proposition  XVII. 


THE  PRE-LOBATCHEIVSKIAN  STRUGGLE        17 

yond  him.  He  starts  from  the  assumption  of  a 
quadrilateral  with  three  right  angles  and  examines 
the  consequences  that  follow  upon  the  hypothesis 
that  the  fourth  angle  is  right,  obtuse,  or  acute.  He 
discovers  that  the  second  and  third  of  these  assump- 
tions are  incompatible  with  the  existence  of  unequal 
similar  figures.  The  second  assumption  gives  the 
triangle's  angle  sum  greater  than  two  right  angles, 
is  incompatible  with  the  theory  of  parallels,  but  is 
realized  in  the  geometry  of  the  sphere.  From  this 
he  was  led  to  conjecture  that  the  third  hypothesis 
might  also  be  realized  on  the  surface  of  a  sphere  of 
imaginary  radius.  This  is  perhaps  the  first  glimpse 
of  the  conception  of  "  pseudospherical  "  surfaces 
afterwards  developed  and  named  by  E.  Beltrami. 
Lambert  also  proves  that  the  departure  of  the 
triangle's  angle  sum  from  two  right  angles  is  a 
quantity  which  is  proportional  to  the  area  of  the 
triangle,  the  larger  the  triangle,  the  greater  the 
departure.  Hence  in  the  case  of  elliptic  and  hyper- 
bolic surfaces  this  angle-sum  is  a  variable  quantity 
which  approaches  two  right  angles  as  the  sides  of 
the  triangle  become  less  and  less.  This  of  course 
points  to  the  fact  that  in  the  infinitesimal  the  spher- 
ical and  pseudo-spherical  triangles  approach  the 
Euclidean  as  a  limit,  and  that  in  endeavoring  to  test 
empirically  the  validity  of  the  various  forms  of 
geometry  for  the  actual  space  world  we  must  seek 


i8       rilL  I'KE-LOBATCHEIVSKIAN  STRUGGLE 

among  very  large  triangles  for  a  measurable  di- 
vergence from  the  Euclidean  conception. 

Another  important  suggestion  which  we  owe  to 
Lambert,  is  that  in  a  space  in  which  the  triangle's 
angle  sum  differs  from  two  right  angles  there  is  an 
absolute  standard  of  measure,  a  natural  unit  of 
length. 

Gauss  and  Lcgendre  also  assumed  that  the  theory 
oi  parallels  is  involved  in  Euclid's  other  assump- 
tions. Gauss  did  not  publish  anything  upon  the 
subject,  and  it  was  not  known  until  after  his  death 
that  he  had  interested  himself  in  it.  His  corre- 
spondence, recently  published,^^  shows  that  he  was 
possibly  in  possession  of  a  non-Euclidean  system, 
but  it  does  not  make  clear  to  what  extent  his  in- 
vestigations were  actually  carried.  He  announced 
in  1/99  that  Euclidean  geometry  would  follow  from 
the  assumption  that  a  triangle  can  be  drawn  greater 
than  any  given  triangle.  In  1804  he  was  still  hop- 
ing to  prove  the  parallel  postulate.  In  1830  he  an- 
nounces that  geometry  is  not  an  a  priori  science. 
In  1 83 1  he  .states  that  non-Euclidean  geometry  is 
non-contradictory,  that  in  it  the  angles  of  the  tri- 
angle diminish  without  limit  when  all  the  sides  are 
increased,  and  that  the  circumference  of  the  circle 
of  radius  r  ~  ttk  (  -  -]    where  k  is  a  constant 

dependent  upon   the  nature  of  space.      It  is  clear 

'■'^  EiiRcl  and  Staeckcl  op.  cit. 


THE  PRE-LOBATCHEIVSKIAN  STRUGGLE        19 

from  this  that  Gauss  had  in  his  possession  the  foun- 
dations of  pseudospherical  or  hyperboHc  geometry, 
and  may  have  been  the  first  to  consider  it  as  prob- 
ably true  ■'■^  of  the  actual  world.  He  came  to  regard 
geometry  merely  as  a  logically  consistent  system  of 
constructs,  with  the  theory  of  parallels  as  a  neces- 
sary axiom ;  he  had  reached  the  conviction  that  this 
"  axiom  "  could  not  be  proved,  though  it  is  known 
by  experience  to  be  approximately  correct.  Deny 
this  axiom  and  there  results  an  independent  geom- 
etry which  he  calls  anti-Euclidean.^^  An  important 
conception  which  Gauss  introduced  was  his  "  Meas- 
ure of  Curvature,"  "**  which  expresses  the  condition 
under  which  a  surface  has  the  property  of  free  mo- 
bility of  figures.  This  "  measure  "  is  the  reciprocal 
of  the  product  of  the  greatest  and  least  radii  of 
curvature  and  remains  unchanged  if  the  surface  is 
bent  without  distension  or  contraction  of  its  parts 
into  any  position.  For  example,  we  can  roll  up  a 
sheet  of  paper  into  the  form  of  a  cylinder  or  cone 
without  changing  the  dimensions  of  figures  drawn 
upon  it,  and  hence  the  geometry  of  the  cylinder  or 
of  the  cone  is  the  same  as  that  of  a  limited  plane. 

In  Legendre's  work  there  is  nothing  new.     His 
results  are  of  interest,  however,  because  of  Dehn's 

^"  Compare  Russell,   Art.   "  Non-Euclidean   Geometry."  The 
New  Encyclopedia  Britannica,  Vol.  XXVIII.,  pp.  674  ff. 
■*o  Gauss  Ztim  Gedacchtiiis.  Leipzig  1856. 
*i  IVerke  Bd.  IV.,  p.  215. 


20      THE  PRE-LOBATCHEWSKIAN  STRUGGLE 

investigation  which  will  come  before  us  later.  Le- 
crendre  was  able  to  prove,  by  assuming  the  mfinity 
or  two-sidedness  of  the  straight  line  and  the  Archi- 
medes Axiom  of  Continuity,  (i)  that  the  sum  of 
the  three  angles  of  a  triangle  can  never  be  greater 
than  two  right  angles,  and  (2)  that  if  in  any 
triangle  this  sum  is  equal  to  two  right  angles,  so 
is  it  in  every  triangle. 


DISCOVERY   AND    DEVELOPMENT 


OF 


NON-EUCLIDEAN  SYSTEMS. 


II 


CHAPTER  11. 

THE    DISCOVERY    AND    DEVELOPMENT    OF    NON- 
EUCLIDEAN    SYSTEMS. 

VVe  pass  now  to  consider  those  attempts  to  solve 
the  riddle  which  proceed  upon  the  hypothesis  that 
if  the  parallel  postulate  is  dependent  upon  Euclid's 
other  assumptions  we  shall  by  denying  it  and  affirm- 
ing them  be  led  into  contradiction.  This  hy- 
pothesis proved  to  be  an  exceedingly  fruitful  one. 
The  absolute  necessity  of  the  parallel  postulate  for 
Euclidean  geometry  and.  the  possibility  of  many 
other  systems  equally  as  rigorous  and  non-contra- 
dictory as  Euclid  itself  have  resulted  from  it. 
Several  independent  discoveries  of  non-Euclidean 
geometry  have  come  to  light.  Schweikart's  was  the 
first  to  be  published ;  ^  Lobatchewsky's  the  first  to 
find  its  way  into  print. - 

1  1812  in  Charkow,  communicated  to  Bessel,  to  Gerling  and 
afterward  to  Gauss  in  1818.  See  Science  N.  S.  Vol.  XII.,  pp. 
842-846. 

"  It  is  interesting  to  note  that  as  late  as  Feb.  25,  i860,  a  non- 
Euclidean  geometry  was  worked  out  by  Prof.  G.  P.  Young  of 
Canada,  entirely  without  knowing  that  anything  of  the  kind 
had  been  previously  done.  See  Canadian  Journal  of  Industry, 
Science  and  Art,  Vol.  V.,  pp.  .341 -.•^S^.  i860. 
23 


jJ^/'2d,       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

Lobatchewsky  2  defines  the  straight  Hne  as  one 
which  fits  upon  itself  in  all  its  positions  so  that  if 
we  turn  the  surface  containing  it  about  two  points 
of  the  line  the  line  does  not  move. 

He  proposes  the  following  substitute  for  Euclid's 
postulate.-*  All  lines  which  go  out  from  a  point  in 
a  plane  may  with  reference  to  a  given  line  in  the 
plane,  be  divided  into  two  classes  —  cutting  and 
non-cut  ting  lines.  If  we  start  in  either  class  and 
move  in  the  direction  of  the  other  we  shall  event- 
ually come  upon  a  line  which  is  the  bounding 
position  between  the  two  classes.  This  line  is  of 
course  unique  and  is  defined  as  being  parallel  to 
the  given  line.  In  the  following  figure,  A  i§  the 
given  point  in  the  plane  and  CD  the  given  straight 
line.  AD  is  perpendicular  to  CD  and  EA  to  AD. 
In  the  uncertainty  wdiether  the  perpendicular  AE 


3  Lobatchewsky  first  communicated  his  results  in  a  lecture 
before  the  Physical  and  Mathematical  faculty  of  the  Univer- 
sity of  Kasan,  of  which  he  was  rector,  in  1826.  They  were 
printed  in  Russian  in  the  Kasan  Messenger  in  1829  and,  in 
ni"ch  more  complete  and  extended  form,  in  the  Gclchrte 
Schriften  der  Unirersitdt  Kasan  1836-1838,  under  the  title 
"  New  Elements  of  Geometry,  with  a  Complete  Theory  of 
PTrallels."  A  briefer  presentation  appeared  in  German  in 
Berlin  in  1840.  Houel  translated  this  into  French  at  the 
suggestion  of  Baltzer  in  1866,  and  there  is  an  excellent  Eng- 
lish translation  by  Halsted,  1896.  The  "  Elements  "  is  by  far 
Lobatchewsky's  greatest  w^ork.  This  and  the  paper  of  1829 
are  now  both  accessible  in  German  in  Prof.  Engel's  vol.,  Leip- 
zig. 1899. 

*  Proposition  16.     Paper  of  1840.  See  Halsted's  Translation. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        25 

is  the  only  line  which  does  not  meet  CD,  we  may 
assume  it  to  be  possible  that  there  are  other  lines, 
for  example  AG,  which  do  not  cut  DC.  how  far 
soever  they  may  be  prolonged.  In  passing  over 
then  from  the  cutting  lines,  as  AF,  to  the  non- 
cutting,  as  AG,  we  must  come  upon  a  line  AH 
parallel  to  DC,  a  boundary  line  upon  one  side  of 
which  all  lines,  AG,  etc.,  are  such  as  do  not  meet 


the  line  DC,  while  upon  the  other  side  every 
straight  line,  AF,.etc..  cuts  the  line  DC.  Now  the 
angle  HAD  between  the  parallel  HA  and  the  per- 
pendicular AD  is  called  the  parallel  angle.  H  this 
be  a  right  angle  the  prolongation  AE'  of  the  perpen- 
dicular will  be  parallel  to  the  prolongation  DB  of 
CD.  In  that  event  every  straight  line  which  goes 
out  from  A,  either  itself  or  its  prolongation,   lies 


j6       UISCOIERY  OF  NON-EUCLIDEAN  SYSTEMS 

in  one  of  the  two  right  angles,  made  by  EE'  upon 
DD'.  which  are  turned  toward  BC,  so  that  all  lines, 
except  the  parallel  EE',  must  intersect  BC  if  they 
are  sufficiently  produced.  In  this  case,  which  is 
Euclid's,  there  is  but  one  line  in  the  plane  parallel 
to  CB.f  But  if  the  "parallel  angle"  be  less  than  a 
right  angle,  and  such  an  assumption  is  perfectly 
legitimate,  there  will  then  lie  upon  the  other  side 
of  AD  another  line  AK  parallel  to  DB  and  making 
DAK  equal  to  the  "  parallel  angle." 

Upon  this  latter  assumption  Lobatchewsky  con- 
structs his  geometry,  proving  in  subsequent  propo- 
sitions that  a  straight  line  maintains  the  character- 
istic of  parallelism  at  every  point,^  that  two  lines 
are  always  mutually  parallel,*'  that  in  a  rectilinear 
triangle  the  sum  of  the  angles  cannot  be  greater 
than  two  right  angles,'  and  that  if  in  any  triangle 
this  sum  is  two  right  angles  the  same  is  true  for  all 
triangles.^  He  styles  this  system  "  Imaginary 
Geometry,"  because  its  trigonometrical  formulae  are 
those  of  the  spherical  triangle  if  its  sides  are  im- 
aginary, or  if  the  radius  of  the  sphere  be  taken 
equal  to  r\^ — i. 

Thus,  by  proving  the  possibility  of  other  systems 
of  geometry.  Lobatchewsky  destroys  the  traditional 

f-  Proposition  XVII. 
8  Proposition  XVIII. 
^  Proposition  XIX. 
*  Proposition  XX. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        27 

trust  in  Euclid  as  absolute  truth,  and  opens  up  a 
vista  of  new  and  suggestive  problems;  nor  was  he 
wholly  unaware  of  the  epistemological  import  of 
his  discovery.  He  remarks,  "  We  cognize  directly 
in  nature  only  motion,  without  which  the  impres- 
sions w^hich  our  senses  receive  are  impossible.  Con- 
sequently all  remaining  ideas,  for  example,  the 
geometric,  are  created  artificially  by  the  mind  since 
they  are  taken  from  the  properties  of  motion,  and 
therefore  space,  in  and  for  itself  alone,  does  not 
exist  for  us." 

John  Bolyai  obtained  results  so  closely  resem- 
bling those  of  Lobatchewsky  that  Russell.  Klein 
and  other  distinguished  writers  have  regarded  the 
tw^o  as  having  had  a  common  inspiration  in  the 
person  of  Gauss.^  History,  however,  does  not  sup- 
port this  conjecture.^  "^  Bolyai's  investigations 
were  published  as  a  twenty-four  page  appendix  to 


»  Russell  does  this  in  his  Foundations  of  Geometry,  p.  8, 
and  Klein,  in  his  Goettingen  lectures  published  in  1893,  p.  175, 
says,  "  Kein  Zzveifel  bestchen  kann,  dass  Lobatcheffsky  so 
luohl  wie  Bolyai  die  Fragestellung  iltrer  untcrsuchttngcn  der 
Gaussischen  Anregung  verdankcn." 

1"  See  Halsted's  Article  in  Science  N.  S.  Vol.  IX.,  pp.  813- 
817,  June  9,  1899.  Schmidt  of  Budapest  has  recently  found  a 
letter  of  Gauss  to  W.  Bolyai  dated  Nov.  25,  1804,  and  accom- 
panied by  a  Latin  treatise  on  parallel  lines.  This  communica- 
tion shows  that  neither  Gauss  nor  Bolyai  had  solved  the 
problem.  Both  believed  the  parallel  postulate  to  be  demon- 
strable and  were  "  racing  to  prove  it." 


28       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

the  "  Tentamcn,"  ^^  a  work  of  his  father,  W. 
Bolyai,  in  1831,  but  its  conception  dates  from  1823. 
He  styles  his  new  geometry  the  "  Science  Absolute 
of  Space."  The  theorems  necessary  and  sufficient 
for  plane  trigonometry  in  this  new  conception  of 
space  are  the  following: 

a'  h' 

( 1 )  sinh-j^=  sin  x"-^"^  ^• 

(2)  cosh  i^  =  cosh    4--     cosh-^,    in    which 

k  k  k 

a'  and  b'  are  the  legs  of  the  right  triangle;  h',  the 
hypothenuse;  A,  the  angle  opposite  a;  and  k,  an 
arbitrary  constant  which  is  presumed  to  be  uniform 
throughout  space.  When  k  is  infinite,  finite  or  im- 
aginary, these  formulas  give  results  which  are  true 
for  Euclidean,  Lobatchewskian.  or  spherical  geom- 
etry respectively.  K  then  is  a  form  of  the  space 
constant. 

Bolyai  shows  a  profounder  appreciation  of  the 
importance  of  the  new  geometry  than  Lobatchew- 
sky.  The  latter  never  explicitly  treats  the  problems 
of  construction  of  the  old  geometry  in  the  changed 
form  which  they  must  take  in  the  new;  such,  for 
example,  as,  "  To  square  the  circle,"  "  To  draw 
through  a  given  point  a  perpendicular  to  a  given 
straight  line,"  and  the  problem  which  in  the  new 


*'  For  full  title  see  Bibliography.  This  work  is  very  rare. 
There  are  only  two  volumes  in  the  United  States.  These  are 
in  possession  of  Professor  Halsted,  to  whom  I  am  indebted 
for  what  knowledge  I  have  of  it. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        29 

geometry  grows  out  of  this :  "  To  draw  to  one  side 
of  an  acute  angle  a  perpendicular  parallel  to  the 
other  side."  All  these  are  elegantly  solved  by 
Bolyai.  He  also  shows  that  the  area  of  the  greatest 
possible  triangle  which,  in  this  new  space,  has  all 
its  sides  parallel  and  its  angles  zero  is  "^i^  where  i  is 
what  we  should  now  call  the  space  constant. 

Like  Lobatchewsky  he  points  out  that  Euclid  is 
but  a  limiting  case  of  his  own  more  general  system ; 
that  geometry  of  very  small  spaces  is  always  Eu- 
clidean; that  no  a  priori  grounds  exist  for  a  deci- 
sion; and  that  observation  can  only  give  an  ap- 
proximate answer  as  to  which  geometry  is  vaild 
for  reality.  Thus  the  new  geometry  casts  no  man- 
ner of  doubt  upon  the  geometry  of  perspective  in 
so  far  as  this  deals  merely  with  incidence  and  coin- 
cidence. Several  propositions  are  equally  true  in 
all  these  geometries,  including  that  of  Riemann, 
which  we  are  next  to  consider.  It  is  mainly  in  the 
measurement  of  distances  and  angles  that  differ- 
ences arise.  In  the  case  of  Euclidean  geometry  the 
infinitely  distant  parts  of  an  unbounded  plane  would 
be  represented  in  perspective  by  a  straight  horizon 
or  vanishing  line,  but  according  to  this  new 
geometry  we  cannot  hold  that  this  line  would  be 
straight;  on  the  contrary  it  would  be  an  hyperbola 
as  in  the  perspective  of  the  terrestrial  horizon.  If 
we  accept  Riemann's  hypothesis  we  cannot  be  sure 
that  there  will  be  any  such  line  at  all.  for  we  do  not 


30       DISCOIERV  OF  NON-EUCLIDEAN  SYSTEMS 

know  that  space  has  any  infinitely  distant  parts.     It 

is  possible  that  if  we  were  to  move  off  in  any  direc- 

,  tion  in  a  straight  line,  we  might  find  that  after 

\  traversing  a  sufficient  distance  we  had  arrived  at 

our  starting  point.  1^  rc'^j^t^e  M<^ '-^i  a.  ,u^\    -  ■^ir' 

It  was  not  the  purpose,  however,  either  of  Lo- 
l)atchewsky  or  of  Bolyai  to  discuss  the  validity  of 
their  own  or  of  Euclidean  geometry.  Their  motive 
was  logical  and  mathematical,  not  epistemological 
or  ontological.  Is  the  result  of  denying  the  parallel 
postulate  contradictory  or  non-contradictory? 
That  was  their  problem.  Nor  did  they  solve  it  com- 
pletely. The  number  of  possible  theorems  in  either 
system  is  practically  infinite.  As  far  as  they  had 
gone  they  were  justified  in  saying  there  were  no 
contradictions,  but  in  nothing  more.  That  latent 
contradictions  might  be  revealed  by  further  de- 
velopments was  perfectly  possible.  For  this  reason 
logical  dependence  or  independence  of  any  group 
of  fundamental  assumptions  can  never  be  com- 
pletely tested  by  the  method  which  they  employed. 
The  purpose  of  Ricmann  and  Hclmholtz  was  a 
'very  different  one.  Their  motive  was  philosophical. 
The  attack  was  no  longer  confined  to  the  parallel 
postulate.  The  problem  was  generalized.  The  old 
synthetic  method  of  Euclid,  still  adhered  to  by 
Lobatchewsky  and  Bolyai.  w'as  now^  abandoned, 
and  the  properties  of  space  were  couched  not  in 
terms  of  intuition,  but  of  algebra.     As  a  result  the 


,    UNIVERSITY   / 

DISCOVERY  OF  NON-EUCLlt&i^&iJ^kS^ffs        31 

subsequent  history  of  non-Euclidean  geometry  took 
on  an  analytical  rather  than  a  synthetic  character. 

Riemann  and  Helmholtz  both  sought  to  show 
that  all  the  so-called  geometrical  axioms  of  Euclid 
are  not  a  priori,  but  empirical  in  character.  In-  his 
most  remarkable  dissertation  ^^  Riemann  expresses 
this  conviction  in  the  following  language :  "  The 
properties  which  distinguish  space  from  other 
triply  extended  magnitudes  are  only  to  be  deduced 
from  experience..  Thus  arises  the  problem  to  dis- 
cover the  simplest  matters  of  fact  from  wliich  the 
measure  relations  of  space  may  be  determined. 
These  matters  of  fact  are,  like  all  mat- 
ters of  fact,  not  necessary  but  only  of  empirical 
certainty." 

Riemann  introduces  into  the  problem  the  gen- 
eral conception  of  a  manifold  of  which  space  is  but 
a  specialization  arrived  at  through  considerations 
of  measurement.  A  manifold  is  continuous  or  dis- 
crete, according  as  there  does  or  does  not  exist 
among  its  specializations  a  continuous  path  from 
one  to  another.  When  in  the  case  of  a  continuous 
manifold  we  can  pass  in  a  definite  way  from  one 
specialization  to  another,  i.  e..  where  continuous 
progress  is  possible  only  forward  or  backward,  the 

^' Die  Hypothesen  zvelchc  dcr  Geometric  su  Gntndc  licgcu 
This  was  his  inaugural  dissertation  before  the  Philosophical 
faculty  of  the  University  of  Goettingen  in  1854.  It  was  not 
published  until  after  his  death  in  1867. 


32       DISfOlERV  OF  NON-EUCLIDEAN  SYSTEMS 

manifold  is  one-dimensional.  If  this  group  of 
specializations  pass  over  into  another  entirely  dif- 
ferent, and  again,  in  a  definite  way  so  that  each 
specialization  of  the  one  passes  into  a  definite  spe- 
cialization of  the  other,  all  the  specializations  thus 
formed  constitute  a  doubly  extended  manifold. 
Similarly  we  may  get  a  triply  extended  manifold, 
by  supposing  a  doubly  extended  one  to  pass  over 
in  a  definite  way  into  another  entirely  different. 
This  process  may  in  general  be  repeated  as  often 
as  we  please,  since  from  the  analytical  point  of  view 
there  is  nothing  to  limit  the  number  of  dimensions.  | 
Space,  however,  as  known  to  us  is  a  manifold  of 
only  three  dimensions  whose  specializations  are 
points.  This  we  know  not  a  priori,  but  as  a  matter 
of  experience.  The  line  is  a  point  aggregate  in 
which  definite  progress  only  forward  or  backward 
is  possible;  a  surface  is  a  one-dimensional  manifold 
of  line  specializations  in  which  progress  always  into 
new  specializations  is  possible  only  forward  or 
backward.  It  is  therefore  a  two  dimensional  point 
aggregate.  In  a  similar  manner  we  obtain  the  solid 
as  a  one-dimensional  surface  aggregate,  a  two 
dimensional  line  aggregate,  or  a  three  dimensional 
point  aggregate.  But  here  according  to  experience 
the  process  stops;  further  progress  is  not  possible, 
for  we  cannot  in  any  way  visualize  the  result. 

Now  to  measure  a  continuous  manifold  certain 
postulates  are  necessary.     Definite  portions  of  it  are 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        zz 

called  Quanta,  and  comparison  of  these  is  accom- 
plished by  measuring,  which  requires  the  possibility 
of  superposing  the  magnitudes  compared.  There- 
fore at  least  one  standard  magnitude  must  be  inde- 
pendent of  position,  i.  e.,  capable  of  being  moved 
about  freely  without  altering  its  value.  Let  us 
suppose  that  it  is  the  length  of  lines  which  is  thus 
independent  of  position  and  that  every  line  is  cap- 
able of  measurement  by  means  of  every  other.  In 
this  way  position  fixing  becomes  a  matter  of  quan- 
tity fixing  and  consequently  the  position  of  a  point 
being  expressed  by  means  of  n  variables  Xi,  Xo,  Xj, 

Xj^,  the  determination  of  the  line  comes  in 

part  to  be  a  matter  of  giving  these  quantities  as 
functions  of  one  variable.  The  problem  then  is  to 
establish  a  mathematical  expression  for  the  length 
of  a  line  and  to  this  end  the  quantities  x  must  be 
regarded  as  expressible  in  terms  of  certain  units. 
To  accomplish  this  let  it  be  assumed  that  the  ele- 
ment of  length  ds  is  unchanged  (to  the  first  order) 
when  all  the  points  undergo  the  same  infinitesimal 
motion;  ds  will  then  become  a  homogeneous  func- 
tion of  the  first  degree  of  the  increments  of  dx  and 
remains  unchanged  when  all  the  dxs,  change  signs. 
The  simplest  case  obviously  is  that  in  which  ds  is 
the  square  root  of  a  quadratic  function  and  this  is 
;he  only  one  which  Riemann  especially  considers. 

We  must  now  consider  Riemann's  conception  of 
:he  "  Measure  of  Curvature  "  of  space  which  is  a 


34       DlSCOl'ERV  OF  S  ON -EL' C  LI  DEAN  SYSTEMS 

somewhat  obscure  and  misleading"  extension  of  the 
Gaussian  conception.     Since  Riemann's  use  of  this 
conception,  especially  as  popularized  by  Helmholtz, 
has  led  to  much  confusion,  it  is  necessary  to  pause 
for  a  moment  and  endeavor  to  understand  what  it 
really  means.     The  conception  goes  back  logically 
as  well  as  historically  to  our  notion  of  the  straight 
line  as  a  measure  of  length.     Since  exact  congru- 
ence   is     essential     to     geometrical     measurement, 
strictly  speaking,  only  a  straight  line  can  be  meas-: 
ured  by  a  straight  line.     If  then  the  stretch  (definite 
portion  of  a  straight  line)  is  to  be  the  standard  of) 
all  linear  measurement,  it  is  evident  that  we  cannotj 
measure  the  circle  except  by  passing  to  the  consid 
eration  of  infinitesimal   arcs,   which   are  to  be  re- 
garded as  straight.     Similarly  our  notion  of  curva- 
ture is  referred  to  the  circle  as  standard  of  meas- 
urement.    The  curvature  of  the  circle  at  any  point 
means,  of  course,  its  amount  of  bending  or  depart- 
ure from  the  tangent  line,  and  since  this  amount  is! 
constant    for  the   same   circle   and   is   equal   to   th(! 
square  of  the  reciprocal  of  the  radius,  it  becomes  eji 
convenient  standard  for  the  measurement  of  lineai 
curvature  in  general.     Now  since  the  curvature  o 
other  curves  varies  from  point  to  point  we  agaii 
pass  to  the  infinitesimal  and  measure  the  amount  o 
this  curvature  by  determining  the  circle  which  mos 
nearly  coincides  with  the  curve  at  the  point  consid 
ered.     This  circle  w^ill  pass  through  three  consecu' 


DISCOIERV  OF  XON-ELCLJDEAX   SyS'lEMS        35 

:ive  ix)ints  of  the  curve  and  hence  its  construction 
s  always  possible  theoretically,^-'  for  any  curve. 
Diane  or  tortuous.  In  an  analogous  way  Gauss 
determined  the  curvature  of  surfaces  by  their 
imount  of  departure  from  the  plane.  In  the  case 
Df  curved  surfaces  we  can  draw  through  any  point 
\n  unlimited  number  of  geodesic  lines  whose  curva- 
;ure,  generally  speaking,  will  not  be  the  same.  If 
:hen  we  draw  all  the  geodesies  from  the  point  at 
ivhich  the  curvature  is  tested  to  the  neighboring 
points  on  the  surface  they  will  form  a  singly  in- 
finite manifold  of  arcs"  among  which  there  will  be 
one  of  maximum  and  one  of  minimum  curvature. 
Representing  the  radii  of  the  osculatory  circles  by 
r  and  ri  we  shall-have  for  the  "  measure  of  curva- 
ture "  of  the  surface  at  the  point  considered  the 
expression  — —  In  case  of  the  sphere,  all  the  geo- 
desies going  out  from  a  point  have  the  same  curva- 
ture, and  hence  the  surface  has  as  its  measure  of 
curvature  Jt-.  For  the  Euclidean  plane  the  radii 
are  all  infinite  and  the  curvature  is  therefore  zero. 
For  the  cone  and  the  cylinder,  since  straight  lines 
can  be  drawn  in  each,  the  measure  of  curvature  is 
zero  and  any  figure  not  too  large  which  can  be 
drawn  upon  the  plane  may  also  be  drawn  without 
distortion  of  parts  upon  the  surface  of  any  cylinder 

13  It   is    interesting  to  note,  however,   that   the   assumption 
that    this   construction    is    possible   is    equivalent   to   assuming 

:he  truth  of  the  parallel  postulate. 


36      DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

or  cone.  It  is  therefore  an  obvious  corollary  that 
for  any  surface  to  possess  the  property  of  free  mo- 
bility its  measure  of  curvature  must  have  a  constant 
value  ^"^  at  every  point. 

But  how  is  it  possible  to  extend  this  conception 
further?  In  what  sense  is  it  permissible  to  speak  of 
the  curvature  of  space?  Thus  far  it  has  been  pos- 
sible to  visualize  results,  and  in  extending  succes- 
sively to  wider  and  wider  applications  the  notion 
of  curvature  we  have  not  departed  at  all  essentially 
from  its  original  meaning.  All  along  it  has  signi- 
fied some  degree  of  departure  from  a  standard  of 
straightness  or  evenness  of  lines  or  of  surfaces. 
Does  the  word  then  still  retain  any  vestige  of  this 
original  meaning  when  applied  by  Riemann  to  the 
conception  of  space?  In  the  consideration  of  sur- 
face curvature  as  given  above,  a  third  dimension 
of  space  was  involved ;  does  the  curvature  then  of 
three  dimensional  space  require  a  fourth  dimen- 
sion? These  are  important  questions,  and  the  fail 
ure  to  comprehend  their  meaning  has  led  to  numer 
ous  vagaries  on  the  part  both  of  mathematician 
and  philosophers. 

In  seeking  an  answer,  let  us  note,  first  of  all,  that 
in  the  above  consideration  of  curvature  as  a  prop- 
erty of  surfaces  no  necessary  reference  is  made  to 
a  third  dimension  in  the  analytic  development  fro 

1*  See  Minding's  proof,  Crelle's  Journal,  Vols.  XIX.,  XX-j 
1839-40. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        3? 

which  it  results.  It  is  only  when  one  tries  to  pic- 
ture to  himself  what  it  may  mean  for  sense  that  he 
seems  to  find  need  of  this  extra  dimension.  Gauss 
assumes  that  points  on  the  surface  may  be  deter- 
mined by  the  two  co-ordinates,  u,  v,  and  finds  that 
small  arcs  of  the  surface  wjU  be  given  by  the 
formula  ( i )  ds-  =  Edu-  +  2Fdu.dv  +  Gdv'-,  in 
which  E,  F,  and  G  are  functions  of  u,  v.  But  u, 
V  may  be  lengths  of  lines  on  the  surface  or  angles 
between  geodesic  lines  and  therefore  we  do  not  need 
to  go  outside  the  surface  itself  for  their  analytical 
determination.  Hence  the  idea  of  a  third  dimen- 
sion may  be  dropped  and  the  measure  of  curvature 
be  analytically  regarded  as  an  inherent  property  of 
the  surface  itself,  provided  of  course  we  are  dealing 
with  a  surface  whose  curvature  is  constant.'^ 

Xow  it  was  this  formula  of  Gauss  which  Rie- 
mann  desired  to  make  use  of  in  determining  a  mean- 
ing for  his  "  measure  of  curvature  "  as  applied  to 
space  of  more  than  tw-o  dimensions.  We  can  see . 
already  how  this  conception  may  be  given  an  analyt- 
ical meaning.  As  in  the  case  of  a  surface  we  find  an 
unlimited  number  of  radii  of  curvature  at  any  point 
corresponding  to  the  geodesies  going  out  from  the 
same,  so  in  regard  to  space  as  a  manifold  of  three 

'■"'  Otherwise  a  precise  co-ordinate  system  which  is  required 
n  the  development  of  our  formula  would  appear  to  be  logic- 
illy  impossible.  Consult  Russell's  Foundations  of  Geometry, 
;hap.  III. 


38       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 


or  of  ;/  dimensions'  we  may  have  an  unlimited  num- 
ber of  measures  of  curvature  at  any  point  corres- 
ponding to  the  surfaces  that  may  be  passed  through 
it.  If  then  we  Hmit  ourselves  to  a  three  dimensional 
manifold  and  represent  its  measure  of  curvature  at 
any  point  by  K  we  shall  have  K=f  (XiX^XsJ  ^15253; 
77,^)where  (X1X2X3)  give  the  position  of  the  point, 
(91S2S3  )  the  orientation  of  the  surface  and  {r  O) 
the  curvature  at  any  point  on  the  surface.  Express- 
ing K  in  terms  of  the  quantities  involved  in  formula 
( I )   for  ds,  we  have, 


(2)K 

I 

1  "[ 

(    i/u 

2v/  E  G  -F  2 

dG' 

d 

+ 

2 

du 

dt> 

LyEG-F 

F             d^ 

] 

El/ 

d7> 

c 

EG-F2 

) 

I F 


Eo/EG-F^ 


rt'E  I 

^7'      V  eg-f 


d¥ 
d%> 


v/'EG-F: 


^E 

dv 


from  which  it_is  j)lain  that  K  may  have  any  value 
depending  upoiTthe  meaning  which  we  give  to  our] 
linear  element  ds.  An  unlimited  number  of  geome- 
tries is  therefore  possible,  the  essential  foundatioi 
of  any  one  of  them  being  the  expression  which  i1 
gives  for  the  distance  between  two  points  takei 
anywhere  and  lying  in  any  direction  from  each^ 
other,  beginning  with  the  interval  between  them  as 
infinitesimal.  If  we  take  ds^rrrdu^-f-dv^  which  \\ 
equivalent  to  assuming  the  existence  of  a  rectangl( 


DISCOVERY  OF  NON-EUCLJDEAX  SYSTEMS        39 

of  which  ds  is  the  diagonal  and  du  and  dv  the  ad- 
jacent sides,  F  in  ( i )  becomes  equal  to  zero  and  E 
and  G  each  equal  to  unity.  Substituting  these  val- 
ues in  (2)  K  becomes  zero,  its  true  value  for 
Euclidean  space.^ 

Riemann's   measure   of  space  curvature   is   then  1 
simply  a  quantity  obtained  by  purely  analytical  cal-  ) 
culation  and  so  far  forth  does  not  necessarily  in-  . 
volve  any  relations  that  would  have  a  meaning  for 
sense-perception.     What  is  the  true  meaning  of  ds? 
That    is    the    important    epistemological    question 
which   his   treatment   suggests,    but   does   not   suf- 
ficiently answer. 

Riemann  himself  works  out  one  form  of  Elliptic 
geometry.  It  is  called  spherical  because  it  holds  for 
the  surface  of  the  sphere.  If  the  measure  of  cur- 
vature has  in  actual  space  a  positive  value  howso- 
ever small  tlrere  are  in  reality  no  such  things  as 
parallel  lines,  for  all  lines  meet  if  sufificiently  pro- 
duced, and  space  itself  is  limited  though  still  un- 
bounded. 

It  is  clear  that  Riemann's  whole  discussion  in- 
volves a  profound  mathematical  and  philosophical 
significance,  the  utmost  reaches  of  which  have  not 
yet  been  fully  explored. 

Helmholtz  and  Riemann  reached  practically  the 
same  conclusions  in  regard  to  the  nature  of  our 
5pace  conception  and  the  empirical  origin  of 
geometric  axioms,  but  their  methods  of  investiga- 


40  DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 
tion  were  quite  different.  Helmholtz  was  a  phys- 
icist and  a  physiologist,  and  was  led  to  consider 
the  problem,  as  he  says,  by  attempting  to  represent 
spatially  the  color  manifold  and  also  by  his  inquiries 
into  the  origin  of  our  ocular  measure  of  distance  m 
the  field  of  vision.  His  method  of  approach  was 
from  the  standpoint  of  natural  science  rather  than 
from  that  of  mathematics  as  in  the  case  of  Rie- 


mann 


While  Riemann  begins  with  an  algebraic  expres- 
sion  which    represents   in   the   most   general   form 
the  distance  between  two  infinitely  near  points  and 
deduces  therefrom  the  conditions  of  the  free  mobil- 
ity of  rigid   figures;   Helmholtz   starts   from   free 
mobility  as  an  observed  fact  in  nature  and  derives 
from  it  the  necessity  of  Riemann's   algebraic  ex- 
pression.    He  also  proves  mathematically  the  lat- 
ter's  arc-formula,   but  his  proof  was   not   strictly 
rigid   and  is  now   superseded  by  that  of   Sophus 
Lie   whose  attention  was  called  to  the  problem  by 
Professor  Klein.     Helmholtz  starts  with  congruence, 
without  which  measurement  is  impossible,  and  ther 
affirms  that  congruence  is  proved  by  experience.^ 
-  We  measure  distance  between  points  by  applying 
to  them  the  rule  or  chain,  we  measure  angles  by 
bringing  the  divided  circle  or  the  theodolite  to  the 
vertex  of  the  angle."     "  Thus  all  geometrical  meas- 

i6"Ueber  die  thatsaechlichen   Grundlagen  der  Geometrie," 
Wissenschaftliche  Abhandhmgen,  Vol.  II..  1866  (p.  610  ff). 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        41 

urement  depends  on  our  instruments  being  really 
as  we  consider  them,  invariable  in  form,  or  at  least 
as  undergoing  no  other  than  the  small  changes  we 
know  of  as  arising  from  variations  of  temperature 
or  from  gravity  acting  differently  at  different 
places."  In  all  our  measuring  then  we  can  only 
make  use  of  the  surest  means  we  have  to  determine 
what  we  are  otherwise  in  the  habit  of  making  out 
by  sight  or  touch.  These  statements  of  Helmholtz 
though  very  suggestive,  do  not  penetrate  to  the 
heart  of  the  matter.  He  is  certainly  right,  however, 
in  starting  as  he  does  with  the  facts  of  experience 
out  of  which  our  geometrical  conceptions,  whatever 
we  may  say  as  to  their  ultimate  source  and  justifi- 
cation, have  come  to  assume  the  forms  in  which 
we  now  have  them. 

It  is  interesting  to  take  note  of  the  axioms  which 
Helmholtz  comes  to  regard  as  being  a  necessary  and 
sufficient  basis  for  geometry.  They  are  briefly  as 
follows :  ( I )  In  a  space  of  //  dimensions,  a  point  is 
uniquely  determined  by  the  measurement  of  //  con- 
tinuous variables  or  co-ordinates. 

(2)  Between  the  211  co-ordinates  of  any  point 
pair  of  a  rigid  body,  there  exists  an  equation  which 
Is  the  same  for  all  congruent  point-pairs.  By  tak- 
ing a  sufficient  number  of  these  point-pairs,  we  can 
E^et  more  equations  than  we  have  unknown  quanti- 
ties, and  thus  the  form  of  the  equations  may  be 
determined  and  all  of  them  satisfied. 


42       DISCOVERY  OF  NON-EUCLlDEAN  SYSTEMS 

(3)  Every  point  can  pass  freely  and  continuously 
from  one  position  to  another.  Hence  by  (2)  and 
(3)  if  two  systems  of  points  m  and  n  can  be 
brought  into  congruence  in  any  position  they  may 
also  be  made  congruent  in  every  other  position, 

(4)  If  (n — i)  points  of  a  body  remain  fixed, 
so  that  every  other  point  can  only  describe  a  certain 
curve  then  that  curve  is  closed. 

If  now  we  limit  ourselves  to  three  dimensions 
Helmholtz  claims  that  these  four  axioms  will  be 
sufficient  to  give  us  the  Euclidean  and  non-Euclid- 
ean systems  as  the  only  alternatives.  This  conclu- 
sion, however,  is  open  to  criticism.  Sophiis  Lie, 
for  instance,  has  shown  ^^  that  the  fourth  axiom  is 
unnecessary.  It  is  included  in  the  axiom  of  con- 
gruence when  properly  formulated.  In  fact,  Con- 
gruence and  Free  Mobility  are  both  involved  in  the 
conception  of  the  homogeneity  of  space.  Russell 
has  also  pointed  out^^  that  these  four  axioms  flow 
directly  from  the  fundamental  assumption  of  the 
relativity  of  position. 

In  a  second  paper/''  which  from  a  mathe- 
matical view-point  constitutes  his  greatest  contribu- 
tion   to    the    subject,    Helmholtz    adds    two    more 


1^  "  Grundlagen  der  Geometric,"  Leipsigcr  Berichte,  1890. 

'■'*"  Foundations  of  Geometry,"  pp.  128  ff,  1897. 

^^ "  Ueber  die  Thatsachen,  die  der  Geometric  zum  Grund 
liegen."  Wissenschaftliche  Abhandlungen,  Vol.  II.,  p.  618  ff., 
1868. 


I 


DISCUrERY  OF  NON-EUCLIDEAN  SYSTEMS        43 

axioms,   namely,   that   space  has  three  dimensions, 
and  that  space  is  infinite. 

The  results  of  Riemann,  Helmholtz,  and  Lie,  had 
in  a  measure  been  anticipated  by  Wolfgang  Bolyai/i 
in  his  famous  "  Tentamen.  "  to  which  we  have 
already  alluded.  Bolyai  starts  his  analysis  with  the 
principle  of  continuity,-'^  then  postulates  the  prin- 
ciples of  congruence  and  free  mobility  of  rigid 
bodies  -^  and  finally  adds  to  these  the  following 
postulates,  (i)  If  any  point  remains  at  rest  any 
region  in  which  it  is,  may  be  moved  about  in 
Innumerable  ways  so  that  any  other  point  than  the 
one  at  rest  may  recur  to  its  former  position.  If 
two  points  are  fixed,  motion  is  still  possible  in  a 
specific  way.  (2)  Three  points  not  co-straight  pre- 
vent all  motion.^-  From  these  assumptions  he  de- 
duces both  Euclid  and  the  non-Euclidean  system  , 
of  his  son.  John  Bolyai.  He  also  observes  that  the 
measurements  of  astronomy  show  that  the  parallel 
postulate  is  not  suffieiently  in  error  to  interfere  in 
practieeP  By  casting  off  the  assum])tion  of  the 
infinity  of  space  Riemann  and  Helmholtz  obtained 
as  a  third  possibility  for  the  universe,  an  elliptic 
geometry;    but    even    this    is    suggested-  by    John 

-^  "  Spatium  est  quantitas,  est  contimiiim."'  P.  44-- 

21  pp.  444,   Sec.   3 :     "  Corpus   idem   in   alio  quoque   loco  vi- 

Jenti   quaestio   succurrit :   mim   loca   siiisdem   divcrsa   aequalia 

unt?"     Intiiitus  ostendit,  aequalia  esse." 
--  p.  446.  Sec.  5- 

23  p.  489. 


44       DlSCOl'ERY  OF  NON-EUCLIDEAN  SYSTEMS 

Bolyai  in  his  proof  that  spherics  are  independent  of 
Euclid's  assumption.^ ^ 

Beltrami  was  the  first  to  become  clearly  conscious 
of  the  fact  that  the  theorems  of  Lobatchewskian 
geometry  may  be  realized  in  ordinary  Euclidean 
space  on  surfaces  of  constant  negative  curvature.^^ 
Minding  had  already  shown^*^  that  the  geometry 
of  such  surfaces  so  far  as  geodesic  triangles 
are  concerned  could  be  deduced  from  that  of  the 
sphere  by  giving  the  radius  an  imaginary  value, 
Beltrami,  however,  generalized  the  problem  first  for 
plane  geometry  in  the  Saggio,  and  in  a  subsequent 
paper  ^^  for  ;/-dimensional  manifolds  of  constant 
negative  curvature.  He  proves  that  a  pseudo- 
spherical  space  of  any  number  of  dimensions  can 
be  considered  as  a  locus  in  Euclidean  space  of 
higher  dimensions.  This  idea  of  space  of  one  type 
as  a  locus  in  space  of  another  type  and  of  dimen- 
sions higher  by  one,  as  Whitehead  says.^'^  is  partly 
due  to  John  Bolyai.  It  is  an  exceedingly  important 
conception  because  it  brings  into  relation  the  geom- 
etries of  Lobatchewsky,  Euclid  and  Riemann,  and 

-*  See  Halsted :  "  Report  on  Progress  of  Non-Euclidean 
Geometry  "  in  Proceedings  of  the  American  Association  for 
the  Advancement  of  Science,  Vol.  48,  1899,  pp.  53-68. 

•  25 "  Saggio  di  Interpretazione  della  Geometria  non-Eu- 
clidea,"  Giornali  di  Matcmatische,  Vol.  VI.,  1868. 

28  Crelle's  Journal,  Vol.  XIX. 

2T "  Teoria  fondamentale  degli  spazzii  di  curvatura  Con- 
stanta "  Aiuiali  di  Matematica  II.,  Vol  II.,  1868-9. 

28  Universal  Algebra,  Cambridge  1898,  Sec  262. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        45 

by  establishing  the  fact  that  demonstrations  of  one 
geometry  may  b^  appropriate  adaptation,  be  made 
to  hold  good  for  corresponding  theorems  in  the 
other  two,  it  afforded,  for  the  first  time,  a  conclusive 
proof  that  the  geometries  of  Lobatchewsky  and 
Riemann  are  no  more  contradictory  than  is  Euclid 
itself,  and  thus  gave  to  all  three  geometries  a  co- 
ordinate rank. 

We  must  now  consider  another  very  striking 
change,  both  as  to  method  and  purpose,  in  the  his- 
torical development  of  Meta-geometry.  Thus  far 
we  have  dealt  altogether  with  metrical  conceptions. 
The  quest  has  been  to  understand  the  necessary  pre- 
conditions to  the  possibility  of  spatial  measurement 
and  to  this  end  space  itself  has  been  regarded  as  a 
species  of  magnitude  whose  peculiar  properties 
need  to  be  defined.  Starting  with  a  doubt  as  to 
the  apodeictic  truth  of  a  single  Euclidean  axiom, 
others  have  been  called  in  question  and  the  convic- 
tion has  grown  that  they  are  all  of  an  empirical 
and  somewhat  arbitrary  character.  One  after  an- 
other of  those  sacred  postulates  which  for  two  thou- 
sand years  even  the  best  minds  had  considered  as 
eternally  true,  was  denied,  and  new  systems  of 
geometry  sprang  into  existence  as  non-contradic- 
tory and,  so  far  as  empirical  observation  could  go, 
as  valid  for  reality  as  Euclid  itself.  Consequently 
that  same  human  desire  for  truth,  logically  pure 
ind  indubitable,  which  for  more  than  twenty  cen- 


46       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

turies  had  reposed  with  perfect  confidence  upon 
Euclid,  gradually  drove  the  geometer  to  seek  the 
longed  for  necessity  and  logical  purity  wholly  out- 
side the  realm  of  metrical  considerations.  Though 
avowedly  mathematical  and  technical  in  its  aims 
and  purposes  this  new  movement  has  attained  cer- 
tain results  that  are  of  far-reaching  philosophical 
importance.  \  Magnitude,  superposition  and  congru- 
ence are  now  dispensed  with  and  the  •  attention  is 
directed  to  the  purely  qualitative,  as  opposed  to 
the  quantitative,  aspects  of  space?, 

Perhaps  the  greatest  names  connected  with  this 
movement  are  those  of  Cayley  and  Klein.  By 
treating  the  surface  of  the  pseudo-sphere  as  a  plane 
and  its  geodesies  (corresponding  to  great  circles  on 
a  sphere)  as  straight  lines  Beltrami  had  shown  that 
all  the  theorems  of  Lobatchewskian  geometry  can 
be  developed  upon  this  surface ;  but  in  Cayley's  new 
Theory  of  Distance  we  seem  to  have  a  much  simpler 
explanation,  and  one  which  requires  no  modification 
of  the  ordinary  conceptions  of  space  or  of  Euclidean 
planes  and  straight  lines,  but  only  an  extension  of 
the  customary  ideas  of  measurement.  Cayley  was 
throughout  a  staunch  defender  of  Euclid. 

In  1858  ^^  he  states  that  "  in  any  system  of  geom- 
etry of  two  dimensions  the  notion  of  distance  can 
be  arrived  at  from  descriptive  principles  alone  by 
means  of  a  conic  called  the  Absolute  and  which  in 
-"  See  Collected  Mathematical  Papers,  Vol.  V.,  p.  550. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        47 

ordinary  geometry  degenerates  into  a  pair  of 
points."  He  sets  himself  the  task  of  estabHshing 
this  position  mathematically  in  his  *'  Sixth  Me- 
moir upon  Quantics"  in  1859.  In  determining  the 
analytic  expression  for  the  distance  of  two  points 
Cayley  first  introduces  the  inverse  sine  or  cosine  of  a 
certain  function  of  the  co-ordinates  and  shows  that 
metrical  properties  become  projective  wath  reference 
to  the  degenerate  conic  called  the  Absolute ;  ^^  but 
later  he  recognizes  and  adopts  Klein's  definition  as 
an  improvement  upon  his  o\vn.^^  This  definition  is 
expressed  by  the  formula :  distance  PO  =  c  log 
AP.BQ 

where   A    and    B    are   the    fixed    points 

AQ.BP 

which  determine  the  /Absolute.  This  formula  pre- 
serves the  fundamental  additive  relation  character- 
istic of  distance;  viz.,  distance  PQ  +  distance  QR 
=  distance  PR. 

We  can  not  here  enter  into  the  details  of  this 
mathematical  discussion  but  must  be  content  with 
the  popular  exposition  which  C^ley  himself  sup- 
plies in  his  Presidential  Address  •"'^  of  1883.  In 
condensed  form  his  conception  is  this.^^  Consider 
an  ordinary  indefinitely  extended  plane;  and  let  us 


^"  Grundgebild,  see  Klein. 

31  Collected  Math.  Papers.  Vol.  II.,  p.  604 

=*-' Collected  Math.  Papers,  Vol.  XL,  pp.  429-459- 

33  Collected  Matl^.  Papers,  Vol.  XI.,  pp.  435  ff. 


48       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

modify  only  the  notion  of  distance.  We  measure 
distance  with  a  foot  rule,  let  us  say.  Imagine  then 
the  length  of  this  rule  constantly  changing  (as  it 
might  do  by  an  alteration  of  temperature)  but 
under  the  condition  that  its  actual  length  shall  de- 
pend only  on  its  situation  in  the  plane  and  on  its 
direction.  In  other  words,  if  its  length  is  a  certain 
amount  for  a  given  situation  and  direction  it  will 
be  the  same  whenever  it  returns  to  this  position,  no 
matter  how,  or  from  what  direction  it  comes.  Now 
it  is  plain  that  the  distance  along  any  given  straight 
or  curved  line  between  any  two  points  could  be 
measured  with  this  rule,  and  always  with  the  same 
determinate  result,  no  matter  from  what  point  in 
the  line  we  begin.  Of  course  this  distance  will  not 
be  what  we  usually  mean  by  the  term,  for  we  do  not 
ordinarily  regard  our  standards  as  varying  quan- 
tities. But  for  aught  we  know  experimentally  this 
may  be  what  actually  occurs.  Suppose  then  that 
as  this  rule  moves  away  from  a  fixed  central  point 
in  the  plane  it  becomes  shorter  and  shorter;  if  this 
shortening  takes  place  with  sufficient  rapidity,  it  is 
clear  that  a  distance  which  in  the  ordinary  sense  of 
the  word  is  finite,  will  in  this  new  sense  be  infinite, 
for  no  number  of  repetitions  of  the  length  of  the 
ever-shortening  rule  will  be  sufficient  to  cover  it. 
There  will  be  then  surrounding  the  central  point  a 
certain  finite  area  every  point  of  whose  boundary 
will  be,  according  to  this  theory,  at  an  infinite  dis- 


DISCO  I  EK  y  OF  NON-E  L  CLIDEA  N  S )  \S- 1  EMS 


49 


tance  from  the  central  point.  Beyond  this  boundary 
there  is  an  unkno^vable  land  or  in  mathematical  lan- 
guage an  imaginary  or  impossible  space. 

By  attaching  to  this  variable  standard  of  Cayley's 
suitable  laws  of  change,  the  various  forms  of  non- 
Euclidean  plane  geometry  may  be  had  upon  this 
Euclidean  plane ;  and  the  idea  may  be  extended  so 
as  to  obtain  non-Euclidean  systems  of  solid  geom- 
etry in  Euclidean  space. 

This  connection  of  Cayley's  Theory  of  Distance 
with  the  various  forms  of  Metageometry  was  first 
pointed  out  by  Felix  Klein.^^  Klein  showed  that 
if  Cayley's  Absolute  be  taken  as  real  we  get  Lo- 
batchewskian,  or  what  he  calls  hyperbolic,  geom- 
etry ;  if  it  be  imaginary  we  get  two  forms  of  elliptic 
geometry;  the  double  elliptic,  spherical,  or  Rieman- 
nian  already  considered,  in  which  all  geodesies  have 
two  points  in  common ;  and  the  single  elliptic  which 
we  owe  to  Klein  ^^  and  in  wdiich  all  geodesies  are 
closed  cur\^es  having  only  one  point  in  common. 
This  is  doubtless  logically  the  simplest  of  all  sys- 
tems. '  Tf  the  Absolute  be  an  imaginary  point  pair 
we  get  parabolic  geometry,  and  if  this  point  pair 
be  what  is  known  as  circular  points  the  result  is 
the  ordinary  Euclidean  system. 

The  natural  result  of  all  this  was  that  both  Cay- 

34  Nicht-EuclicI,  Bonk  T..  Chapters  I.  and  II. 

35  Also  independently  discovered  by  Simon  Newcomb.     See 
his  article  in  Crelle's  Journal,  Vol.  8,3. 


50       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

ley  and  Klein  came  to  regard  the  whole  question 
of  non-Euclidean  systems  as  having  no  philosoph- 
ical importance,  since  it  seemed  to  them  that  it  in 
no  way  concerns  the  nature  of  space,  but  only  the 
definition  of  distance,  which  in  their  view  is  per- 
fectly arbitrary,  being  merely  a  question  of  conven- 
ience. 

Whether  we  accept  this  conclusion  or  not  it  must 
be  admitted  that  the  projective  method,  employed 
by  them,  is  independent  of  metrical  presuppositions 
and  deals  directly  with  that  qualitative  likeness  of 
geometrical  figures  which  is  a  necessary  prerequisite 
to  quantitative  comparison.  The  distinction  be- 
tween Euclidean  and  non-Euclidean  geometry  is  a 
metrical  one;  it  essentially  disappears  altogether  in 
projective  geometry.  Hence  projective  geometry 
deals  with  the  conception  of  space  from  a  higher 
point  of  view,  which  includes  within  its  scope  every 
variety  of  metrical  space  and  whose  defining  adjec- 
tives must  in  consequence  possess  very  great  philo- 
sophical interest.  Furthermore,  these  adjectives 
may  also  be  regarded  as  the  simplest  indispensable 
requisites  of  geometrical  reasoning. 

In  this  connection  the  important  work  of  Sophus 
Lie,  which  was  awarded  the  Lobatchewsky  prize 
Nov.  3,  1897.  must  briefly  claim  attention. ^^  Felix 
Klein   declared  that  this   work   excelled   all   others 

,ia  "  Theorie  der  Transformations  griippen,"  Vol.  III.  Pub- 
lished at  Leipzig,  1893. 


DISCOVERY  OF  NON-EiCLlDEAS  SYSTEMS        51 

SO  absolutely  that  no  possible  doubt  could  be  enter- 
tained as  to  the  justice  of  this  award.  Helmholtz 
had  already  originated  the  idea  of  studying  the 
essential  characteristics  of  space  by  a-  consideration 
of  the  movements  possible  in  it.  Klein  called  the 
attention  of  Lie  to  this  problem  of  Helmholtz  and 
encouraged  him  to  undertake  an  investigation  of 
it  by  means  of  his  Theory  of  Groups.  We  can  but 
meagerly  indicate  the  outcome  of  this  investigation. 
As  stated  by  Lie  his  problem  is :  "  To  determine  all 
finite  continuous  groups  of  transformations  in  three 
dimensional  space  in  which  two  points  have  a  single 
invariant  and  more  than  two  points  have  no  essen- 
tial invariant";  meaning  by  invariant  the  distance. 
D,  between  the  two  points  and  by  the  statement 
that  more  than  two  points  have  no  essential  in- 
variant, no  invariant  which  is  not  expressible  in 
terms  of  D.  He  finds  that  under  the  conditions  of 
the  problem  the  group  must  be  six-parametered  and 
transitive  and  cannot  contain  two  infinitesimal 
transformations  whose  path  curves  coincide.'*" 
Two  solutions  of  the  problem  are  given.  He  first 
investigates  a  group  in  space,  possessing  free  mobil- 
ity in  the  infinitesimal  in  the  sense  that  if  a  point 
and  any  line  element  through  it  be  fixed,  continuous 
motion  shall  still  be  possible;  but  if.  in  addition, 
any  surface  element  through  this  point  and  the  line 

3^  Transformations-Gnippcn.  \'ol.  III.,  p.  405  f.,  1893. 


52        DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

element  be  fixed,  no  continuous  motion  shall  be 
possible.  The  groups  which  in  tri-dimensional 
space  harmonize  with  these  conditions  Lie  finds  to 
be  only  those  which  are  characteristic  of  the  Euclid- 
ean and  non-Euclidean  geometries,  but  strange  to 
say  he  also  discovers  that  for  the  apparently  an- 
alogous but  simpler  case  of  the  plane  or  two  dimen- 
sional space  there  are,  besides  these,  certain  other 
groups  where  the  paths  of  the  infinitesimal  trans- 
formations are  spirals.  In  his  second  demonstra- 
tion, starting  from  transformation-equations  with 
Helmholtz's  first  three  postulates  he  proves  that  for 
a  space  of  three  dimensions  the  fourth  postulate  is 
entirely  superfluous.^^ 

From  an  analytical  point  of  view  Professor  Hil- 
bert  in  a  recent  article  in  the  Mathematische  Anna- 
len,^^  which  may  be  mentioned  here,  has  advanced 
beyond  these  results  of  Lie  by  showing  that  it  is 
possible  to  do  away  with  the  dififerentiability  of 
functions  which  Lie's  discussion  requires.  From 
the  intuitional  standpoint  his  article  ofifers  no  im- 
provement and  is  open  to  certain  criticisms  which 
Dr.  Wilson  ""^  has  pointed  out. 

•■'^  For  a  brief  statement  of  what  is  essentially  Lie's  method 
in  English,  see  Halsted's  "  Columbus  Report,"  Proc.  A.  A.  A. 
S.,  Vol.  48,  1899.  Also  Poincare's  Art.  in  Nature  previously 
cited. 

■■'s  Bd.  56,  Heft  3,  pp.  381-422,  October,  1902. 

*f>  Archiv  der  Mathematik  und  Physik  III.  Rcihe  VI.  1.  u.  2. 
Heft,  Jan.  1903. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        53 

In  his  famous  Festschrift,^^  however,  Professor 
Hilbert  has  done  more  perhaps  than  any  one  else 
except  certain  ItaHans  to  determine  the  precise  num- 
ber, meaning  and  relations  of  the  postulates  essential 
to  geometry.  In  this  older  work  Hilbert  followed 
essentially  the  Euclidean  method  with  a  logic  so 
keen  and  pure  and  a  result  so  simple  that  many 
have  even  expressed  the  opinion  that  it  will  ulti- 
mately supersede  Euclid  in  the  elementary  schools. 
Certain  defects,  however,  have  been  pointed  out  by 
Schur,"*-  Moore, ■^-^  and  others,  showing  that  Hil- 
bert's  postulates  are  not  independent  as  he  had  sup- 
posed they  were,  and  also  illustrating  how  difficult 
it  is  to  satisfy  logic  when  one  seeks  to  determine 
the  foundations  of  geometry  by  the  intuitional 
method.  The  discovery  of  this  fact  has  led  Hilbert 
in  his  recent  article  to  abandon  this  method  for  the 
more  strictly  logical  one.  He  starts  from  the  ideas 
of  Manifoldncsses  and  Groups  as  Lie  had  done,  but 
uses  the  new  conception  of  Manifoldncsses  intro- 
duced by  Georg  Cantor  thus  dispensing  with  any 
special  reference  to  a  system  of  co-ordinates  in  a 
geometric  space. 

When  the  Lobatchewsky  prize  was  awarded  to 
Lie.  the  thesis  of  M.  L.  Gerard,  of  Lyons,  also  re- 

*i  Grundlagen  der  Geometric,  Leipzig  1899. 
*2  Mathematische  Annalcn  Bd.  55.  p.  265  flF. 
*3  Transactions  of  the  American  Mathematical  Societj-.  Vol. 
III.,  pp.  142  ff. 


54       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

ceived  honorable  mention.  In  this  thesis  Gerard 
endeavors  to  estabhsh  the  fundamental  propositions 
of  non-Euclidean  geometry  without  any  hypothet- 
ical constructions  except  the  two  which  are  assumed 
by  Euclid."*-^  ( i )  Through  any  two  points  a 
straight  line  can  be  drawn.  (2)  A  circle  may  be 
described  about  any  center  with  any  given  sect  as 
radius.  But  in  order  to  establish  the  relations  be- 
tween the  elements  of  a  triangle  in  a  thorough- 
going manner  he  adds  to  these,  two  other  assump- 
tions as  follows :  ( I )  A  straight  line  which  inter- 
sects the  perimeter  of  a  polygon  in  some  other  point 
than  one  of  its  vertices  intersects  it  again,  and  (2) 
two  straight  lines,  or  two  circles,  or  a  straight  line 
and  a  circle,  intersect  if  there  are  points  of  one  on 
both  sides  of  the  other.  One  of  the  most  important 
considerations  for  the  advocates  of  non-Euclidean 
geometry  is  the  requirement  that  all  its  figures  shall 
be  rigorously  constructed.  It  was  to  meet  this  re- 
quirement that  Gerard's  investigation  was  under- 
taken and  it  is  in  this  fact  that  its  significance 
mainly  lies. 

When  the  Commission  of  the  Physico-Mathe- 
matical  Society  of  Kazan  met  in  1900  for  the  pur- 
pose of  awarding  again  the  Lobatchewsky  prize, 
they  found  before  them  two  new  treatises  on  non- 

■**  TTiis  idea  was  suggested  and  partially  developed  by  G. 
Battaglini  in  his  "  Sulla  Geometria  Imaginaria  di  Lobatchew- 
sky," Giornale  di  Mat.  Anno  V.,  pp.  217-231,  1867. 


DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS        55 

Euclidean  geometry,  the  merits  of  which  were  so 
nearly  equal  that  the  decision  between  them  was 
finally  made  by  the  casting  of  lots.  These  were 
A.  N.  Whitehead's  investigations  in  his  "  Universal 
Algebra  "  "^^  and  Wilhelm  Killing's  **  Grundlagen 
der  Geometric."  '**^ 

In  the  opinion  of  Sir  Robert  Ball,  Whitehead's 
investigation  excels  anything  previously  done  in 
two  important  particulars.  In  the  first  place  he  can 
treat  n-dimensions  by  practically  the  same  formulae 
as  those  used  for  two  or  three  dimensions;  and 
secondly,  the  various  kinds  of  space,  parabolic, 
hyperbolic  and  elliptic,  present  themselves  in  White- 
head's methods  quite  naturally  in  the  course  of  the 
work,  where  they  appear  as  the  only  alternatives 
under  certain  definite  assumptions.  Perhaps  the 
most  significant  portion  of  Killing's  effort  is  his 
treatment  of  the  "  Clifford-Klein  space- forms." 
whose  importance  lies  in  the  fact  that  they  show 
what  a  difference  it  makes  whether  we  assume  the 
validity  of  our  fundamental  axioms  for  space  as  a 
whole  or  only  for  a  completely  bounded  portion  of 
space.  The  first  assumption  yields  the  Euclidean 
and  three  non-Euclidean  space-forms  already  men- 
tioned, but  the  second   gives  a  "  manifoldness.  at 


*^  Cambridge,  England,  I? 
4«  Paderborn,  1898. 


56       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

present    not    yet    dominated,     of    different    space 
forms."  ^'' 

We  must  now  call  attention  to  the  remarkable 
results  of  Max  Dehn's  investigation,  which  was 
undertaken  at  the  suggestion  of  Professor  Hilbert 
and  published  in  1900.^^ 

As  early  as  1898  Friedrich  Schur  had  reached 
the  conviction  that  elementary  geometry  can  be  built 
up  without  the  use  of  the  Archimedes  axiom  of 
!  continuity, ''^  and  proved  Pascal's  theorem  without 
^the  use  either  of  this  axiom  or  of  the  parallel 
postulate.  He  constructs  a  sect  Calculus  in  which 
he  shows  that  the  theory  of  proportion  can  be 
founded  without  the  introduction  of  irrational 
numbers  and  indicates  that  this  might  also  be  done 
without  the  Archimedes  Axiom.  Hilbert  accom- 
plished this  proof  in  1899  and  demonstrated  thai 
this  axiom  need  no  longer  be  regarded  as  necessar} 
to  elementary  geometry.  As  we  have  already  stated 
Legendre,  by  assuming  this  axiom  and  also  tha' 
the  straight  line  is  of  infinite  length,  demonstratec 
( I )  that  the  angle  sum  of  any  plane  triangle  canno 
be  greater  than  two  right  angles,  and  (2)  that  if  ii 


*'  From  Professor  Engel  of  Leipzig,  in  a  Russian  pamphle 
printed  at  Kazan,  taken  from  Halsted's  translation. 

*8  Dehn  was  a  pupil  of  Hilbert.  He  was  21  years  old  whei; 
this  investigation  was  completed.  It  is  printed  in  Mat.  Ami 
53  Band,  pp.  404-439- 

*^  See  Preface  to  his  Lehrbuch  der  Analytischen  Geometii 
Leipzig,  1898. 


i 


DISCOVERY  OF  NON-EUCLIDEAN  SySTEMS        5; 

any  triangle  this  sum  is  equal  to  two  right  angles, 
the  same  is  true  of  every  triangle.  Hence  the  ques- 
tion arose,  Do  these  two  theorems  actually  hold 
good  in  Euclidean  geometry?  And  the  problem 
suggested  for  Dehn  was,  Can  these  theorems  of 
Legendre  be  proved  without  the  Archimedes 
Axiom  ?  ^^  The  results  of  his  investigation  are  very 
remarkable.  He  demonstrates  the  second  of  Le- 
gendre's  theorems  without  any  postulate  of  continu- 
ity, and  shows  that  the  first  theorem  cannot  be  dem- 
onstrated without  the  Archimedes  Axiom.  This  is 
done  by  constructing  a  new  geometry  in  which  an 
infinite  number  of  lines  can  be  drawn  through  a 
point  parallel  to  a  given  straight  line,  but  in  which 
also  the  triangle's  angle  sum  is  greater  than  two 
right  angles.  By  assuming  the  Archimedes  Axiom 
and  also  that  an  infinity  of  parallels  can  be  drawn  to 
a  given  straight  line,  through  a  given  point.  Lo- 
batchewsky's  geometry,  in  t\^hich  the  triangle's  angle 
sum  is  less  than  two  right  angles,  results ;  but  Dehn 
now  shows  that  by  denying  the  Archimedes  Axiom 


50  This  axiom  as  stated  by  Dehn  at  the  beginning  of  his 
thesis  is  as  follows :  If  A^  be  any  point  upon  a  straight  h'ne 
between  any  given  points,  A  and  B,  then  we  can  construct  the 
points  A„  A.J  A^  ...  so  that  Aj  lies  between  A  and 
A^,  A,  between  A^  and  A^,  A,  between  A„  and  A^  and  so 
forth; "and  moreover  the  sects  AA^,  AjA^,  A,A,.  A^.\^. 
.  .  .  are  equal  to  each  other;  then  there  is  in  the  scries 
of  points  A,  A^  A^  .  .  .  a  point  A^  such  that  B  lies 
between  A  and  An- 


58       DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

this  angle  sum  is  either  greater  than  two  right 
angles  or  else  equal  to  two  right  angles,  it  cannot 
be  less  than  two.  He  proves  the  former,  as  just 
stated,  by  his  non-Legendrian  geometry,  and  the 
latter  case,  namely,  that  this  angle  sum  is  equal  to 
two  right  angles,  by  constructing  another  geometry 
in  which  the  parallel  postulate  does  not  hold,  but  in 
which  nevertheless  all  the  theorems  of  Euclid  are 
shown  still  to  be  true.  He  proves  that  the  sum  of 
the  angles  of  the  triangle  is  two  right  angles,  and 
that  various  other  theorems  previously  held  to  be 
exactly  equivalent  to  the  parallel  postulate  are  still 
valid  in  this  new  geometry  in  which  the  parallel 
postulate  is  thus  contradicted. 

His  results  are  well  summarized  by  the  following 
table : 


The     an- 
gle    sum 
in  the  tri- 
angle   is: 

Through  a   given  point   we  can   draw  to  a 
straight : 

No    parallel. 

One  parallel. 

An   infinity  of 
parallels. 

>2R 

=  2R   ^ 
<2R 

Elliptic 
geometry 

(Impossible) 
(Impossible) 

(Impossible) 

Euclidean 
geometry 

(Impossible) 

Non-Legendrian 
geometry 

Semi-Euclidean 
geometry 
Hyperbolic 
geometry 

If  these  results  of  Dehn  should  withstand  future 
criticism  and  prove  to  be  logically  unimpeachable 


DISCOyERV  OF  NON-EUCLIDEAN  SySTEMS      59 

they  will  vindicate  Euclid  in  a  remarkable  manner, 
for  not  one  of  the  proposed  substitutes  for  his 
parallel  postulate  is,  after  all,  its  exact  equivalent, 
except  when  the  axiom  of  Archimedes  is  already 
assumed. 

It  is  impossible  in  this  brief  historical  survey  to 
do  justice  to  certain  important  contributions  that 
have  very  recently  appeared  and  which  from  differ- 

V 

ent  pomts  of  startmg  have  throwrT-:ji  new  light  upon 
the  foundations  of  mathematics  in  general.  We 
refer  to  the  contributions  of  Dedekind/'^  Cantor, 
Peano,  Fieri,  Padoa,  Poincare,  Vailati,  Russell. 
Frege,  and  others  on  number,  continuity,  series,  and 
other  topics  which  come  to  be  involved  in  any  thor- 
ough consideration  of  the  fundamentals  of  descrip- 
tive, projective,  and  metrical  geometry.  Collectively 
considered  these  contributions  reveal  a  very  decided 
movement  to  carry  the  whole  of  so-called  pure 
mathematics  over  to  a  final  grounding  in  formal  or 
symbolic  logic.  We  shall  refer  to  certain  features 
of  this  movement  in  subsequent  chapters. 

It  now  remains  to  notice  with  a  word,  in  closing 
this  chapter,  the  efforts  that  have  been  made  to  deal 
with  the  philosophical  problems  created  by  meta- 
geometry.     These  efforts  have  proceeded  sometimes 


SI  Some  of  these  writings  are  in  reality  not  so  recent  as  some 
of  those  which  we  have  already  considered,  but  they  all  belong 
to  the  one  general  movement  to  which  we  wish  to  call  at- 
tention. 


6o      DISCOVERY  OF  NON-EUCLIDEAN  SYSTEMS 

from  the  mathematicians  themselves,  as  in  the  case 
of  Riemann  and  Helmholtz,  and  sometimes  from 
students  of  philosophy.  With  one  notable  excep- 
tion they  have  all  suffered  more  or  less  from  the 
writer's  inability  to  take  the  point  of  view  of  the 
philosopher  on  the  one  hand  or  of  the  mathematician 
on  the  other  —  an  incapacity  due  to  lack  of  special 
training,  to  see  the  problem  clearly  and  steadily  in 
both  its  mathematical  and  its  philosophical  relations. 
The  exception  referred  to  is  that  found  in  the  con- 
tributions of  B.  A.  W.  Russell,  especially  in  his 
Principles  of  Mathematics,^^  the  first  volume  of 
which  only  recently  appeared.  Mr.  Russell  brings 
to  his  study  of  the  problem  the  training  both  of  a 
mathematician  and  a  philosopher ;  the  result  is  a  con- 
tribution of  remarkable  and  permanent  value. 

In  view  of  the  historical  development  thus  some- 
what imperfectly  presented,  we  shall  endeavor  in  the 
chapters  which  follow,  ( i )  to  orient  the  problem 
and  point  out  its  complex  relations;  (2)  to  trace 
the  parallel  postulate  and  its  closely  allied  concep- 
tions to  their  psychological  sources;  (3)  to  deter- 
mine the  nature  and  validity  of  this  postulate  and  its 
place  in  geometrical  systems;  and  finally  (4)  to  in- 
dicate the  conclusions  which  seem  to  follov/  from 
this  discussion  as  to  the  nature  of  space. 

"'-  Published  at  Cambridge,  England,  1903.  This  is  beyond 
question  the  best  work  on  the  Philosophy  of  Mathematics  yet 
published. 


GENERAL  ORIENTATION 


OF   THE 


PROBLEM. 


CHAPTER  III. 

A   GENERAL   ORIENTATION    OF   THE   PROBLEM. 

The  foregoing  historical  sketch  brings  promi- 
nently to  view  certain  in^tters  of  great  philosophical 
interest^  First  of  all  it  has  certainly  become  clear 
that  in  so  far,  at  least,  as  any  system  of  geometry 
has  professed  scientific  value,  or  has  claimed  to  be 
in  any  sense  valid  for  reality,  the  whole  history  of 
meta-geometry  has  been,  as  a  matter  of  fact,  one 
long  and  very  fruitful  search  for  the  philosophic 
foundations  of  mathematics  in  general.  -The  same 
spirit  which  through  the  centuries  endeavored  so 
earnestly  to  justify  Euclid  as  an  orderly  system  of 
necessary  and  indubitable  knowledge  by  removing 
the  objectionable  theory  of  parallel  lines,  has  finally 
subjected  the  fundamentals  of  arithmetic  as  well  as 
those  of  geometry,  to  the  most  searching  critical 
testing.  The  sufficiency,  independence,  and  mutual 
compatibility  of  the  various  adjectives  which  pre- 
sumably define  our  notions  of  number  and  space. 
have  become  problems  of  absorbing  interest  and 
promise.  As  is  usual  in  every  marked  intellectual 
advance,  every  existing  difficulty  removed  has 
63 


64  ORIENT  AT  ION  OF  THE  PROBLEM 

opened  up  new  fields  of  research,  new  tendencies  of 
thought  and  methods  of  investigation,  and  conse- 
quently new  and  more  difficult  problems  calling  for 
solution. 

The  light  thus  thrown  both  directly  and  indirectly 
upon  the  space  problem  has  led  to  a  very  great  re- 
finement of  the  space  conception  which  has  resulted 
more  and  more  in  restricting  the  a  priori  realm  and 
in  handing  over  to  the  empirical,  as  possibly  contin- 
gent and  depending  ultimately  upon  the  peculiar 
nature  of  experience,  certain  matters  which  were 
previously  thought  to  be  apodeictically  true.  Prior 
to  Lobatchewsky  "  geometry  upon  the  plane  at  in- 
finity "  was  considered  as  being  just  as  well  known 
as  the  geometry  of  any  portion  of  the  table  upon 
which  I  am  writing,  but  today  the  geometer  "  knows 
nothing  about  the  nature  of  actually  existing  space 
at  an  infinite  distance;  he  knows  nothing  about  the 
properties  of  this  present  space  in  a  past  or  a  future 
eternity."  ^  He  does  know,  however,  that,  within 
the  limits  of  the  utmost  refinements  of  instrumenta- 
tion and  observation  thus  far  attained,  the  assump- 
tions of  Euclid  are  true  for  small  portions  of  space 
and  perhaps,  when  all  due  allowance  for  probable 
error  is  made,  even  for  that  immense  region  which 

1  W.  K.  Clifford :  Lectures  and  Essays,  Vol.  I.,  p.  359.  Lon- 
don, 1901.  We  do  not  subscribe  to  the  naive  space-realism 
latent  in  these  words  of  Clifford ;  the  passage  is  quoted  because 
it  indicates  very  clearly  the  changed  point  of  view  regarding 
the  nature  of  geometry. 


ORIENTATION  OF  THE  PROBLEM  65 

is  swept  by  telescopic  vision.  Hence  the  important 
question  as  to  what  are  the  necessary  and  sufticient 
marks  of  the  category  of  space,  once  regarded  as 
settled,  takes  on  decidedly  a  new  interest  for  specu- 
lative thought. 

Glancing  back  for  a  moment  over  the  history  of 
this  movement,  one  can  easily  trace  from  a  psycho- 
logical view-point  the  predominant  intellectual  and 
practical  interests  out  of  which  it  has  grown.  It  is 
often  contended  that  geometry  is  concerned  with 
ideal  objects.  At  present,  this  is  certainly  true;  but 
as  a  matter  of  history  it  has  not  always  been  true. 
Geometry  is  in  reality  a  complex  product  of  two 
factors;  the  one  empirical  or,  if  you  please,  intu- 
itional, and  the  other  logical.  Both  appear  to  be 
necessary  to  any  geometry  which  would  validate  its 
claims  to  be  a  bona  fide  body  of  systematic  knowl- 
edge. Our  historical  sketch  shows  that  these  factors 
have  been  variable  quantities,  the  intuitional  element 
having  been  steadily  reduced  until  at  present  so  far 
as  space  is  concerned  it  is  entirely  rejected.  Geom- 
etry as  a  part  of  "pure"  mathematics  is  coming  to 
be  regarded  as  merely  a  branch  of  symbolic  Logic, 
which  no  longer  claims  to  throw  direct  light  upon 
the  nature  of  space. 

Xevertheless,  in  contemplating  the  "  pure  "  ab- 
stract science  which  thus  claims  to  be  free  from  all 
intuitional  bias,  we  should  not  forget  its  humbler 


1« 


66  ORIENTATION  OF  THE  PROBLEM 

origin.  Even  among  the  Egyptians  ^  and  the  early 
Greeks,^  where  authentic  history  first  finds  the  sub- 
ject already  somewhat  advanced,  it  is  almost  wholly 
an  empirical  matter.  In  the  hands  of  the  later 
Greeks,  however,  the  treatment  of  geometry  under- 
went essential  modifications.  This  naive  conception 
of  things  disappears.  Geometry  passes  from  a 
purely  technical  to  a  scientific  state  and  becomes 
the  subject  of  professional  and  scholarly  contempla- 
tion. For  the  first  time  a  conscious  effort  is  made 
to  separate  the  directly  cognizable  from  what  is 
logically  deducible  and  to  throw  into  distinct  relief" 
the  thread  of  deduction.  For  purposes  of  instruc- 
tion the  principles  which  are  simplest,  most  easily 
gained  and  apparently  freest  from  doubt  and  contra- 
diction are  placed  at  the  beginning,  and  the  re- 
mainder based  ypon  them.  The  motive  now  arises 
to  reduce  the  number  of  these  principles  as  far  as 
possible.  In  this  respect  the  superiority  of  Euclid 
was  recognized,  as  we  have  seen,  for  more  than 
twenty  centuries.  His  selection  of  fundamental  con- 
ceptions seemed  to  withstand  every  opposition  and 
all  efforts  at  a  further  reduction. 

Having  at  length  found,  however,  that  the  denial 
of  Euclid's  parallel  postulate  led  to  a  different  sys- 

-  Compare  Eisenlohr :  Bin  Mathematischen  Handbuch  dcr 
alten  Aegypter:    Papyrus  Rhind,  Leipzig,  1877. 

^  James  Gow :  A  Short  History  of  Greek  Mathematics. 
Cambridge,  1884. 


ORIENTATION  OF  THE  PROBLEM  67 

tern  which  was  self-consistent  and  possibly  true  of 
the  actual  world,  a  new  motive  arose.  "  Mathe- 
maticians "*  became  interested  in  developing  the  con- 
sequences flowing  from  other  sets  of  axioms  more 
or  less  resembling  Euclid's.  Hence  arose  a  large 
number  of  geometries  inconsistent  as  a  rule  with 
each  other  but  each  internally  consistent."  Even 
the  resemblance  to  Euclid  at  first  required  in  any  set 
of  axioms  which  it  was  desired  to  investigate  was 
gradually  disregarded.  Possible  systems  were  in- 
vestigated on  their  own  account,  and  thus,  as  we 
have  said,  the  intuitional  aspect  of  geometry  became 
altogether  a  matter  of  indifference.  Geometrical 
propositions  are  no  longer  assertorical  in  character. 
They  do  not  claim  to  state  what  actually  is,  but! 
merely  assert  that  certain  consequences  flow  from  1 
given  premises.  Whereas  Euclid  asserted  not  only 
that  certain  geometrical  inferences  were  logically 
sound  but  also  that  both  the  premises  and  the  con- 
clusion were  actually  true,  the  new  geometry  pro- 
nounces upon  the  inference  merely,  and  leaves 
premises  and  conclusion  both  as  matters  of  doubt. 
The  implications  alone  belong  to  geometry;  with 
axioms  and  propositions  it  is  not  concerned.  The 
geometer  deals  with  certain  entities  to  be  sure,  but 
these  are  carefully  defined  and  guaranteed  to  exist 
only  in  the  sense  that  they  are  logically  compatible. 

♦  Hon.    B.    A.    W.    Russell's    "  Principles    of    Mathematics," 
Vol.  I.,  p.  373.     Cambridge,  1903. 


68 


ORIENTAriON  OF  THE  PROBLEM 


They  are  not  even  necessarily  points,  or  lines,  or 
any  of  those  objects  usually  regarded  as  the  legiti- 
mate subject-matter  of  geometry,  but  may  be  any] 
mental  constructs  which  harmonize  with  certain  con- 
ditions   arbitrarily    chosen.     The    question    as    toj 
whether  any  set  of  axioms  and  propositions  hold  of] 
actual  space  or  not,  is  then  a  problem  of  appliec 
mathematics,   to  be  decided,   so   far  as  decision  ii 
possible,    by    experiment    and    observation.     Purf 
mathematics  contents   itself  with  merely  asserting 
that  if  any  space  has  such  and  such  properties  it  wil 
also  have  such  and  such  other  properties. 

Riemann's  generalization  through  the  intro 
tion  of  analytical  conceptions  so  extensively  employee 
by  subsequent  writers  and  the  important  works 
Dedekind,^  Cantor,^  and  others,  on  the  nature  o! 
continuity,  have  given  rise  to  new  interests  in  lin< 
with  the  general  demand  for  logical  rigor  and  havj 
shown  the  necessity  of  subjecting  the  prerequisite 
of  analytical  geometry  to  a  careful  investigation. 

In  the  employment  of  the  analytical  method  space 
was  regarded  as  a  manifold  of  points  referred  to 
system  of  co-ordinates  and  each  capable  of  bein^ 
definitely  determined  by  means  of  numbers.     Line 
were  defined  by  establishing  a  one  to  one  correspond- 
ence between  the  ensemble  of  numbers  and  a  certaii 


5  "  Was  sind  und  was  sollen  die  Zahlen."     Brunswick,  1893.1 
6 "  Ein     Beitrag    zur     Mannigfaltigkeitslehre,"     Borchardt'sl 
Journal.  Band  84,  pp.  242-258,  Dec,  1877. 


ORIENTATION  Of  THE  I'KOBLEM  69 

series  of  points;  surfaces,  by  setting  up  a  similar 
correspondence  between  the  ensemble  of  numbers 
and  a  series  of  lines;  solids,  by  establishing  the  same 
correspondence  between  the  ensemble  of  numbers 
and  a  certain  series  of  surfaces.  Thus  the  geometri- 
cal continuum  came  to  be  regarded  as  generated  by 
the  number  of  continuum,  and  the  question  naturally 
arose  as  to  how  this  course  of  procedure  may  be 
justified.  Dare  we  include  all  numbers  in  the  en- 
semble spoken  of,  imaginary  and  real,  and  rational 
and  irrational?  In  the  development  of  geometry 
this  has  actually  been  done.  It  is  indeed  "  impos- 
sible to  exaggerate  the  importance  even  of  imaginary 
numbers,  for  without  them  the  fabric  of  modern 
geometry  could  not  stand  for  a  moment."*  Hence 
in  investigating  the  relations  of  Euclid  to  the  mod- 
ern system,  especially  if  we  regard  the  latter  as  the 
more  ultimate  and  "  pure,"  the  question  of  right 
becomes  an  interesting  one.  Is  there  not  really, 
though  perhaps  not  so  patently,  as  much  room  for 
doubt  here  as  in  the  case  of  the  parallel  postulate? 
Given  a  co-ordinate  system,  we  may  readily  admit 
that  if  any  set  of  quantities  actually  determine  a 
point,  they  determine  it  uniquely,  but  hoiv  do  ur 
know  that  they  determine  it  at  all?  That  is  the 
interesting  question.  To  say  that  to  each  of  a  cer- 
tain series  of  objects  there  corresponds  a  number  is 

'Professor  Edwin  S.  Crawley.  Popular  Sci.  Mon.,  Jan..  lOOi. 


70  ORIENTATION  OF  THE  PROBLEM 

surely  quite  different  from  saying  that  to  every  pos- 
sible number  there  corresponds  an  object  in  the  given 
series.  Cayley's  Absolute  requires  for  Euclidean 
geometry  circular  points  at  infinity.  Dare  we  as- 
sume that  there  is  anything  in  reality  corresponding 
to  these  mythical  entities?  Not  pausing  to  reply 
but  only  to  indicate  the  general  course  of  this  inter- 
esting development,  we  need  only  say  that  if  this 
question  be  answered  affirmatively  there  arises  an- 
other no  less  interesting  and  difficult,  which  has 
contributed  its  share  to  the  quest  for  mathematical 
rigor  resulting  in  the  general  thought  movement 
characterized  by  Professors  Pierpont  ^  and  Klein  as 
the  "  Arithmetization  of  Mathematics."  Rational 
numbers  apparently  give  no  trouble.  Their  arith- 
metic is  easy,  but  when  it  comes  to  interpolating 
among  these  that  infinity  of  irrational  numbers,  such 
as  radicals,  logarithms  and  others,  met  with  in  the 
development  of  mathematics,  it  has  been  customary 
to  proceed  upon  the  tacit  assumption  that  the  arith- 
metic of  these  is  the  same  as  that  of  rational  num- 
bers and  that  whatever  operations  may  be  performed 
upon  the  one  class  may  also  be  performed  upon  the 
other.  Thus  the  Kantian  ''  Quid  Juris? "  again 
confronts  us.  Certain  eminent  mathematicians  have 
replied  that  no  rigor  is  possible  except  upon  a  basis 
of  rational  numbers. 

8  Bull.  Amer.  Math.  Soc,  2d  series,  Vol.  V.,  No.  8,  pp.  379- 
385.     May,  1899. 


ORlENTATIOy  OF  THE  I'KOBLEM  71 

To  meet  the  demands  of  such  reasoning  the  old 
Aristotehan  logic  was,  of  course,  not  altogether  ade- 
quate, it,  too,  like  Euclid,  stood  in  need  of  critical 
renovation  and  extension.  To  get  rid  of  the  want 
of  accuracy  which  creeps  in  unnoticed  through  the 
association  of  ideas  and  is  therefore  not  allowed  for 
when  ordinary  language  is  employed,  symbols  for 
different  logical  processes  were  introduced.  It  was 
found  that  all  deduction  is  not  syllogistic  as  the 
scholastics  had  thought.  Asyllogistic  inferences 
must  also  be  recognized.  A  new  logic  was  created, 
for  which  Boole,  C.  S.  Peirce,  and  Peano  have 
been  largely  responsible.  To  this  final  court  of  ap- 
peal all  mathematical  difiiculties  are  henceforth  to 
be  brought.  Russell's  latest  work  is  an  elaborate 
and  thorough-going  effort  to  establish  the  thesis  that 
all  pure  mathematics,  geometry  included,  is  merely 
a  branch  of  this  symbolic  logic. 

All  this  then,  as  inspired  by  one  supreme  motive, 
an  age-long  struggle  for  w^hat  men  have  seen  fit  to 
call  absolute  rigor.  If  we  turn  to  the  cautious 
mathematician  and  ask  what  he  means  by  this  word, 
how  he  shall  know^  when  he  has  attained  it.  and 
what  is  his  standard,  we  do  not  find  him  at  all  ready 
to  reply.  Indeed  it  is  wise  to  be  silent,  for  much 
that  was  once  thought  to  be  rigorous  is  now  no 
longer  so  regarded;  a  large  part  of  the  reasoning 
of  the  last  century  would  be  rejected  today.  An 
English  or   American  treatise  on   Calculus  twenty 


^2  ORIENTATION  OF  THE  PROBLEM 

years  old  is  now  almost  as  obsolete  as  a  work  on 
chemistry  of  the  same  date  would  be.  Geometrical 
rigor  is  a  variable  quantity  approaching  a  limit 
which  can  scarcely  be  reached  except  as  mathematics 
becomes  wholly  divorced  from  actual  sense  experi- 
ence. But  geometry  refuses  to  be  thus  a  mere  -mat- 
ter of  logic.  So  regarded  its  territory  can  only  be 
arbitrarily  defined.  It  becomes  a  mere  study  of 
multiple  series,  so  Russell  has  actually  defined  it, 
and  as  such  it  includes  complex  numbers  as  a  legiti- 
mate part  of  its  subject-matter.  But  why  restrict 
it  to  multiple  series,  why  not  also  include  series  of 
only  one  dimension,  and  thus  do  away  with  the  name 
geometry  altogether.  Indeed  it  is  only  when  we 
introduce  some  notion  of  applied  geometry,  some 
conception  so  defined  as  to  resemble  more  or  less 
approximately  what  we  know  to  be  true  of  our 
actual  space,  that  our  employment  of  this  term 
seems  to  be  anything  more  than  an  unjustifiable  mis- 
use of  words.  Geometry  as  logic  may  indeed  care 
very  little  as  to  the  particularized  existence,  either 
actual  or  possible,  of  the  entities  with  which  it  deals, 
but  geometry,  as  geometry,  in  any  justifiable  mean- 
ing of  this  word,  is  certainly  something  more  than 
this.  And  even  as  pure  logic  it  is  in  a  sense  still 
subject  to  the  space  category;  its  entities  are  con- 
ceived as  delimited,  externalized,  and  otherwise 
spatially  related.  They  are  supposed  to  be  capable 
of  certain  spatial  transformations.     How,  then,  do 


I 


I 


ORIENTATION  OF  THE  PROBLEM 


73 


we  know  that  certain  positions  and  transformations 
of  these  entities  are  allowable,  except  as  we  fall  back 
upon  those  more  ultimate  axioms  which  though  they 
have  a  wider  application  are  still  in  an  important 
sense  geometrical.  They  are  logical  necessities 
which  cannot  even  be  thought,  or  regarded  as  hav- 
ing any  meaning  whatever,  without  reference  to 
some  sort  of  co-existent  realities  actual  or  abstractly 
possible  which  are  conceived  of  as  entering  into  re- 
lations with  each  other  which  are  only  partially 
defined  by  these  "axioms "  and  which  inevitably 
imply  the  space  category. 

But  upon  the  consideration  of  the  logical  and 
epistemological  questions  thus  brought  to  view  it  is 
not  our  purpose  at  present  to  enter.  The  aim  here 
is  simply  to  point  out  the  motive  dominating  tills 
movement  and  to  show  that  the  general  problem 
which  lies  before  us  is  one  of  extreme  complexity, 
which  has  been  by  no  means  generally  understood 
by  those  who  have  essayed  its  solution. 

Certainly  the  profound  questions  growing  out  of 
the  parallel  postulate,  its  validity  for  reality,  tiie 
general  nature  of  space  which  this  involves  and  the 
fundamental  relations  of  Euclid  to  other  systems  'if 
geometry  can  never  be  settled  by  pure  mathematics 
alone,  remarkable  as  the  contributions  fmm  this 
source  have  certainly  been.  The  non-Knclidcnn 
movement,  as  such,  has  proliably  produced  already 
rlmost  all  the  modifications  it  is  likely  to  produce  in 


74  ORIENTATION  OP  THE  PROBLEM 

the  foundations  of  geometry.'-^  In  the  final  answer, 
if  indeed  a  final  answer  is  possible,  the  psychologist, 
the  philosopher,  the  physicist,  and  the  physiologist 
must  each  have  something  to  say.  The  claims  of 
individual  investigators  to  have  solved  the  problem 
completely,  and  such  claims  even  yet  occasionally 
appear,  can  only  be  looked  upon  with  a  measure  of 
suspicion. 

On  the  philosophical  side  of  the  problem  the  old 
question  as  to  the  a  priori  nature  of  the  postulates  of 
Euclid  affirmed  by  Kant  and  denied  by  Mill,  is  still 
in  debate.  The  psychological  analysis  necessary  to 
a  just  and  fruitful  treatment  of  this  question  has 
at  times  been  entirely  wanting ;  at  others,  its  signifi- 
cance has  been  utterly  ignored.  The  term  a  priori 
itself  has  been  used  in  widely  dififerent  senses  by 
different  writers  and  sometimes  even  by  the  same 
writer.  The  word  "  curvature,"  as  applied  to  space, 
unfortunately  introduced  by  Riemann,  Helmholtz, 
and  Beltrami,  has  also  been  a  source  of  perennial 
confusion  for  both  mathematicians  and  philosophers 
alike.  The  essential  meaning  of  the  straight  line 
as  used  in  the  various  forms  of  geometry,  and  of  the 
subordinate  conceptions  of  distance,  direction,  and 
motion  by  which  this  line  is  usually  defined,  has  not 
been  clearly  determined.  And  finally  the  abstract 
spaces  of  geometry  demand  further  study  as  regards 

^  Consult  Russell's  ."  Principles  of  Mathematics,"  p.  381. 


ORIENTATION  OF  THE  PROBLEM  75 

their  relations  to  each  other  and  to  the  space  of 
actual  experience.  They  require  to  be  more  thor- 
oughly tested  by  that  conception  which  must  arise 
when  all  bona  iide  human  experiences  of  a  spatial 
order  are  taken  into  the  account. 

In  the  next  chapter  we  enter  upon  a  psychological 
study  of  the  parallel  postulate  and  certain  closely 
allied  conceptions  without  which  this  postulate 
would  have  no  meaning  at  all.  Our  purpose  will 
be  tw^ofold.  First,  starting  with  experience,  we 
shall  endeavor  to  trace  the  genesis  and  development 
of  these  conceptions  as  they  appeal  to  the  ordinary 
consciousness  of  man;  and  secondly,  starting  with 
these  conceptions  in  the  highly  abstract  forms  de- 
manded by  modern  geometry,  we  shall  try  to  see 
how  far  these  forms  may  be  made  to  have  meaning 
for  actual  experience. 


I 


THE    PSYCHOLOGY  OF  THE  PAR 
ALLEL  POSTULATE 

AND    ITS 

KINDRED  CONCEPTIONS. 


CHAPTER  IV. 

PSYCHOLOGICAL   SOURCES  OF   THE  PARALLEL  POSTU- 
LATE   AND    ITS    CLOSELY   ALLIED    CONCEPTIONS. 

There  can  be  little  doubt  that  geometry  sjjrang 
originally  from  man's  interest  in  the  spatial  rela- 
tions of  physical  bodies.  It  bears  in  every  part  un- 
mistakable evidence  of  this  empirical  origin,  and 
the  course  of  its  development  can  be  rendered  fully 
intelligible  only  on  consideration  of  these  traces. 
Various  forms  of  sensory  experience  contributed 
the  data.  By  virtue,  it  may  be.  of  the  peculiar 
structure  of  the  body  with  its  pairs  of  sense-organs 
symmetrically  located,  whose  conscious  deliverances 
possess  in  each  case  a  remarkable  similarity  as  con- 
trasted wfth  those  of  other  sense-organs,  the  power 
of  orienting  these  organs  themselves  and  finally  of 
the  whole  body  with  reference  to  presented  stimuli, 
has  at  last  been  acquired.^     This  power  of  orienta- 

1  Consult  Loeb's  Physiology  of  the  Brain  (N.  Y.,  lOOoV 
Also  Royce's  Outlines  of  Psychology,  pp.  I39-I47-  It  '«  P*"''- 
haps  due  to  this  cause  that  the  lower  animals  know  how  to 
strike  in  and  to  hold  more  or  less  steadily  a  straipht  direction 
in  movement.  Consult  Von  Cyon's  articles  in  Pflncger's 
Archiv   fner  Physiologie. 

79 


8o  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

tion,  taken  with  sensations  of  movement,^  sight,  and 
touch,^  and  of  the  so-called  statical  sense  of  the  semi- 
circular canals,  furnishes  the  empirical  basis  for  the 
perception  of  space  as  a  continuous  whole. 

The  unitary  perception  of  space  thus  arising  is, 
of  course,  complex  in  its  nature  and  is  determined 
in  each  case  by  the  character  of  the  sensory  factors 
which  it  actually  involves.  So-called  psychological 
spaces  corresponding  to  different  senses  are  not 
wholly  identical.  The  space-perception  of  a  man 
born  blind  ^  is  unitary  in  character,  but  quite  differ- 
ent from  that  of  a  man  whose  vision  remains  un- 
impaired. These  sensory  differences  have  been  un- 
consciously carried  over  into  the  foundations  of 
geometry,  so  that  the  different  forms  of  this  science 
can  be  classified  as  motor,  visual,  etc.,  according  as 
special  emphasis  has  been  laid  in  their  construction, 
now  on  one,  now  on  another,  of  these  sensory  fac- 
tors. Projective  geometry  is  entirely  visual,  while 
Euclid  is  largely  motor.  These  psychological  dis- 
tinctions are  important  when  questions  of  validity ' 
are  raised  regarding  the  foundations  of  competitive 
systems.     Had  man's  spatial  experience  been  con- 

-  Poincare  holds  that  without  sensations  of  movement  and 
the  actual  ability  to  move  geometry  could  never  arise.  The 
Monist.  See  also  Professor  Ladd's  A  Theory  of  Reality,  pp. 
229-230. 

3  Ladd's  Psychology  Descriptive  and  Explanatory,  pp.  323  f. 

*  Consult  Dr.  Alexander  Cameron's  Thesis  on  "  Tactual  Per- 
ception."   Yale,  1900. 


il 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM       8i 

fined,  for  example,  to  vision'^  alone,  the  struggle 
between  Euclid  and  Lobatchewsky  could  never  have 
been,  since  for  vision  alone  there  are  no  such  things 
as  parallel  lines.  Through  a  point  in  a  plane  there 
is  neitiier  one  nor  a  pencil  of  lines  which  do  not  cut 
a  given  line  in  the  plane ;  every  such  line  is  seen  to 
converge  toward  the  given  line  in  one  or  in  both 
directions. 

When,  therefore,  we  come  to  inquire  what  are 
the  sensory  factors  that  enter  into  the  subordinate 
conceptions  which  define  the  spaces  o'f  the  different 
geometries,  where  in  each  the  emphasis  is  actually 
laid,  in  the  choice  of  conceptions  and  in  the  charac- 
ter of  the  demonstrations  which  follow  upon  them, 
and  finally  what  is  the  true  balance  to  be  maintained 
among  these  sensory  factors  in  determining  the  na- 
ture of  space  as  it  appears  to  be  actualized  in  the 
world  of  reality,  it  is  plain  that  the  psychological 
difficulties  involved  strike  deeper  than  mere  ques- 
tions of  accommodation  and  convergence  in  visual 
perception.  Nevertheless,  investigations  of  this  sort 
have  thrown,  and  no  doubt  will  continue  to  throw, 
light  upon  the  general  problem.  We  organize  our 
experiences  to  harmonize  with  our  own  and  with 
the  movements  of  physical  bodies;  consequently 
those  changes  in  the  environment  of  an  object  which 
necessitate  changes  in  the  character  of  the  move- 

5  This  is  perhaps  an  impossible  hypothesis,  but  it  scn-es  our 
purpose  in  this  connection. 


82   PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

ments  by  which  it  is  perceived  occasion  differences 
in  the  perception  of  its  spatial  properties.  In  this 
way  optical  illusions  arise.  With  these  so-called 
false  perceptions,  eye-movements  ^  are  now  known  to 
be  closely  related  if  not  indeed  a  determining  cause. 
The  fact  that  long-continued  practice  dispels  these 
illusions,  the  perceived  object  remaining  constant 
while  the  perception  itself  gradually  but  uncon- 
sciously changes,  indicates  that  the  peculiar  marks 
of  any  concept  of  space  founded  simply  upon  visual 
perceptions  can  hardly  be  called  a  priori  in  Kant's 
sense  of  the  word.  Photographs  taken  at  intervals 
during  the  presence  of  these  optical  illusions  and 
after  they  have  finally  disappeared  show  quite 
clearly  that  changes  in  eye-movements  correspond- 
ing to  those  in  the  perception  itself  successively  oc- 
cur. Increasing  accuracy  of  movement  and  correct- 
ness of  perception  develop  together.  Whether  the 
movement  or  the  perception  itself  is  to  be  regarded 
as  the  determining  cause  of  this  improvement  is  an 
interesting  but  difficult  question.  It  seems  very 
probable,  however,  that  cerebral  organization  and 
accurate  motor  adjustment  must  first  be  secured 
before  correct  perception  becomes  at  all  possible. 
]f  the  mere  recognition  of  error  were  all  that  is 
needed,  no  long-continued  process  of  perceptual  edu- 

•^  Based  upon  some  very  interesting  but  as  yet  unpublished 
experiments  in  photographing  eye-movements  in  the  Yale 
Laboratory  by  Assistant  Professor  Judd  and  Dr.  McAllister. 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM        s-. 

cation  would  seem  to  be  required.  Correct  percep- 
tion would  then  take  place  as  quickly  and  completely 
as  when,  through  a  false  perception,  we  have  mis- 
taken some  stranger  for  a  friend.  ^^ 

Turning  now  from  these  general  considerations 
to  study  those  special  facts  from  which  geometry 
as  a  science  has  actually  developed,  we  lind  that  the 
first  geometrical  knowledge  which  can  be  strictly 
so-called  was  acquired  accidentally  and  without  de- 
sign through  practical  experiences  in  varied  employ- 
ments. This  peculiar  empirical  origin  is  shown  in 
W'hat  history  w^e  have  of  the  beginnings  of  geometry 
among  the  ancient  Egyptians  "^  when  as  yet  the  scien- 
tific spirit  in  its  search  for  the  logical  interconnec- 
tions of  the  experiences  in  question  had  not  arisen. 
It  appears  also  even  more  clearly  in  the  history  of 
primitive  civilizations  at  large,  as  may  be  seen  in 
the  rise  of  metrical  conceptions  and  in  those  facts  of 
experience  out  of  which  the  conception  of  the 
straight  line  and  Euclid's  theory  of  parallels  have 
developed.  We  shall  now  study  these  developments 
in  the  order  here  mentioned. 

Primitive  man  was  already  well  advanced  in  cer- 
tain geometrical  ideas  before  measurement,  strictly 
so-called,  began.  He  had  acquired  a  considerable 
practical  knowledge  of  physical  bodies  and  their 
grosser  spatial  relations  without  taking  advantage 

7  Consult  Gow's  A  Short  History  of  Greek  M.ithematirs. 
Cambridge,  1884. 


84  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

of  this  artificial  assistance.  Through  the  compari- 
son of  various  kinds  of  sensory  experiences  he  had 
come  to  attribute  to  bodies  a  certain  spatial  con- 
stancy; he  had  learned  to  locate  them,  to  estimate 
their  form,  size,  and  distance,  with  considerable 
accuracy  and  to  govern  his  actions  accordingly.  But 
it  was  in  the  objects  themselves,  in  their  capacity 
to  satisfy  his  needs  and  not  in  their  spatial  relations 
that  his  interests  primarily  centered.  It  was  only 
when  he  had  so  far  triumphed  over  his  foes  in  the 
struggle  for  self-preservation  as  to  be  able  to  reflect 
that  the  place,  form,  and  size  of  objects  have  almost 
everything  to  do  with  determining  the  character  of 
those  activities  by  which  his  wants  are  best  satisfied 
and  his  enemies  overcome  or  avoided,  that  the  desire 
for  a  more  accurate  determination  of  these  quanti- 
ties by  means  of  measurement  arose.  His  first  esti- 
mates were,  of  course,  obtained  by  the  comparison 
of  memory  images  with  present  perceptions.  This 
mode  of  estimation,  however,  depends  upon  certain 
physiological  and  psychological  conditions  which 
are  difficult  to  control  and  is  therefore  unsatisfac- 
tory when  exact  measurement  is  required.  This  is 
especially  true  when  the  interval  of  time  between  the 
experiences  compared  is  large  and  the  memory  image 
has,  as  a  consequence,  considerably  faded.  Hence 
it  becomes  necessary  to  provide  characteristics  which 
depend  as  little  as  possible  upon  these  conditions,  and 
this  can  only  be  done  by  removing  the  time  interval 


PSYCHOLOGIC  .ISPBCTS  OF  Tllli  PROBLEM       85 

between  the  remembered  and  the  perceived  experi- 
ences altogether,  or  by  rendering  it  as  brief  as  pos- 
sible,—  the  ideal  being  the  substitution  of  direct 
perception  in  the  place  of  memory.  .  This  can  l)c 
done  only  by  securing  the  exact  congruence  of  the 
-bodies  compared,  and  hence  it  is  always  theoretically 
impossible,  because  of  the  inevitable  limitations  of 
sense-perception,  and  the  want  of  absolute  rigidity 
in  all  natural  objects.  Nevertheless  this  is  the  prin- 
ciple of  measurement,  and  it  remains  the  same  for  all 
spatial  measurements  whether  i)er formed  by  the  low- 
est savage  or  the  most  exact  geometer,  as  a  direct 
perception  of  physical  congruence  or  as  a  purely  ab- 
stract visualization. 

For  all  physical  measurement,  then,  a  convenient 
portable  standard  of  some  sort  is  needed,  and  one 
whose  want  of  appreciable  variation  during  trans- 
portation may  be  directly  perceived.  Naturally  the 
first  objects  of  this  sort  to  appeal  to  the  primitive 
man  would  be  various  parts  of  his  own  body.  The 
names  of  the  oldest  measures,  and  various  other 
facts,  indicate  clearly  that  these  were  the  standards 
actually  employed.,^  Among  these  names,  for  exam- 
ple, are  the  hand,  nail,  ell,  span,  cubit,  foot,  fathom, 
pace,  and  mile.  These  names  have,  of  course,  lost 
much  of  their  original  meaning,  and  have  now  come 
to  be  associated  with  standard  measures  which  they 
happen  to  represent  with  only  tolerable  accuracy. 

It  w^as  a  significant  forward  step  in  civilization 


86  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

when  nations  like  the  Egyptians  and  Babylonians 
abandoned  these  physiological  for  more  exact  phys- 
ical standards.  But  the  evolution  of  this  form  of 
measurement  has  also  taken  place  in  accordance 
with  known  psychological  laws.  In  harmony  with 
the  general  principle  that  it  is  in  the  material  rather 
than  in  the  more  distinctly  spatial  properties  of  ob- 
jects that  human  interests  first  become  centered  the 
first  measurements  of  this  sort  were  doubtless  meas- 
urements of  volume,^  not  measurements  by  means 
of  definitely  chosen  standards  of  volume,  however, 
but  merely  inaccurate  estimates  of  the  capacity  of 
vessels  and  storerooms  by  determining  the  quantity 
or  number  of  similarly  shaped  bodies  which  they 
would  contain.^  Similarly  the  first  estimates  of 
area  were  probably  made  by  the  number  of  fruit- 
bearing  trees  which  a  field  would  grow,  the  amount 

s  It  is  interesting  to  note  that  those  who  have  given  most 
attention  to  the  best  methods  of  introducing  geometry  to  be- 
ginners have  generally  agreed  that  it  is  best  to  begin  with 
solids  and  considerations  of  volume,  rather  than  with  points, 
lines,  and  angles,  and  there  is  a  decided  movement  now  on  in 
elementary  education  in  America  to  follow  with  the  individual 
a  method  very  similar  to  that  by  which  as  we  here  find  the  race 
has  been  educated.  Note  some  recent  American  publications 
such  as  Campbell's  Observational  Geometry,  New  York,  1899; 
Spear's  Advanced  Arithmetic,  Boston,  1899;  Hanus's  Geometry 
in  the  Grammar  School,  Boston,  1898.  Tlie  Harvard  Catalogue 
for  1901-1902,  p.  307:  Row's  Geometric  Exercises  in  Paper- 
Folding,  Chicago,  Open  Court,  1901 — Tr. 

^  E.  Mach :  The  Development  of  Geometry,  TTie  Monist, 
Vol.  XII.,  p.  486.  Also  Eisenlohr:  Papyrus  Rhind,  etc.,  pre- 
viously cited. 


PSYCHOLOGIC  ASPECTS  OP  THE  PROBLEM       87 

of  labor  necessary  to  cultivate  it.  or  the  number  of 
animals  that  could  be  grazed  upon  it.  The  meas- 
urement of  one  surface  by  another  may  have  been 
suggested  by  estimating  in  this  way  the  relative 
value  of  fields  which  lay  near  one  another. 

Herodotus  ^°  states  that  when  Xerxes  wished  to 
count  the  army  which  he  led  against  the  Greeks  he 
adopted  the  device  of  drawing  up  10.000  menxlosely 
packed  together  within  an  enclosure  which  was  then 
made  to  serve-  as  a  standard,  and  each  succeeding 
division  that  filled  it  was  counted  as  another  10,000. 
This  is  an  example  of  another  type  of  operations 
which  naturally  led  to  the  measurement  of  one  sur- 
face by  another.  By  abstracting  at  first  instinct- 
ively and  then  consciously  from  the  height  of  the 
practically  identical  bodies  thus  covering  a  surface 
the  notion  of  a  surface  unit  w^ould  finally  be  reached. 
The  fact  that  among  the  Egyptians.^'  the  early 
Greeks,  and  even  as  late  as  the  Roman  '-  sTrrreyo^u- 
rules  for  the  measurements  of  surfaces  of  irregular 
figure  were  often  grossly  inaccurate,  seems  in  gen- 
eral to  favor  this  view\^'' 

But  what  needs  to  be  emphasized  most  perhaps  in 

1"  Herodotus  VII.,  22,  56,  103,  223. 

^^  Eisenlohr :  op.  cit. 

^-' M.  Cantor:     Die  romischcn  Agriiiicnsoren.  Leipzif?.  1875. 

"  Thucydides  VI.,  i  states  that  surfaces  havinR  equal  per- 
imeters have  equal  areas.  And  Ahmcs  in  Papyrus  Rhind  Rcts 
the  area  of  the  triangle  by  multiplying  together  two  of  its 
sides. 


88  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

connection  with  measurement  is  the  fact  that  all  acts 
of  measurement  exhibit  the  free-mobility  of  approxi- 
mately rigid  bodies,  and  that  these  facts  when  con- 
ceptualized bring  to  clear  consciousness  the  corre- 
sponding postulate  which  lies  at  the  foundation  of 
every  system  of  metrical  geometry.  And  further- 
more the  recognized  possibility  of  constructing  simi- 
lar solids  of  different  sizes  leads  by  a  similar  path- 
way to  the  postulate  of  homogeneity. 

The  modern  method  of  defining  points,  lines,  and 
surfaces  as  boundary  conceptions,  though  logically 
necessary  and  presenting  little  difficulty  to  a  mind 
skilled  in  abstract  thinking,  nevertheless  conceals 
rather  than  exhibits  the  process  whereby  these  con- 
ceptions have  developed.  The  straight  line  still 
bears  in  its  name  "a  suggestion  of  its  origin. 
Straight  is  the  participle  of  the  old  verb  to  stretch 
and  line  is  from  line^t,  which  signifies  a  thread,  hence 
the  straight  line  literally  means  a  stretched  cord  or ; 
thread.  By  decreasing  the  thickness  of  such  an  ob- 
ject until  it  becomes  vanishingly  small  the  concep- 
tion of  the  line  as  a  geometrical  magnitude  of  only 
one  dimension  is  reached  by  an  easy  abstraction. 

By  making  fast  one  end  of  a  string  and  drawing 
the  other  through  a  hole  or  a  ring  and  observing 
how  more  and  more  of  it  passes  out  until  the  whole 
becomes  stretched  or  straight  we  have  an  example 
of  that  type  of  experiences  from  which  the  notion  of 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM       89 

the  straight  Hne  as  the  shortest  distance  between 
two  poyits  came  to  be  clearly  discerned. 

Another  class  of  simple  experiences  supplies  that 
peculiar  property  of  the  straight  line  by  which  it  may 
be  defined  as  a  unique  figure  determined  by  any  two 
of  its  points.  If  we  slide  a  crooked  material  line 
betw'een  any  two  fixed  points  it  will  be  observed 
that  the  portion  between  the  two  points  will  con- 
tinually alter  as  regards  the  form  and  position  of 
its  parts.  The  more  uniform  the  object,  the  less 
this  variation  becomes  until  the  limit  of  perceptual 
uniformity  is  reached/ when  the  object  will  be  seen 
to  slide  within  itself.  We  now  have  a  property 
which  belongs  obviously  as  much  to  the  circle  or  the 
__S£iral  as  it  does  to  the  straight  line,  for  these  figures 
also  when  thus  operated  upon  are  seen  to  possess  it. 
But  if  we  rotate  these  objects  about  the  two  points 
in  question  a  difference  is  at  once  detected,  and  we 
come  upon  a  peculiar  property  of  the  straight  line. 
It  rotates  unthin  itself. 

Now^  when  we  consider  these  two  properties  of 
the  straight  line.  tJwt  it  rotates  unthin  itself  ami  is 
also  the  shortest  distance  hetnren  t7i'o  points,  it  be- 
comes perfectly  obvious  why  this  figure  has  been 
made  fundamental  in  all  systems  of  geometry.  // 
is  the  only  one-dimensional  magnitude  that  is  physio- 
logically simple  and  perceptually  constant  when 
vieived  from  any  point  not  in  it.  The  same  is  true 
of  the  plane,  and  explains  its  unique  position  among 


90  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

two  dimensional  magnitudes.  It  is  the  perceptual 
simplicity  and  constancy  of  these  two  figures  that 
has  forced  them  upon  us  as  the  only  invariable,  and 
hence  the  only  satisfactory,  metrical  standards. 
Among  tridimensional  objects  the  cube  and  the 
sphere  hold  a  similar  relation,  and  for  a  similar 
reason.  Of  these  two  figures  the  sphere  is  of  course 
the  simpler  from  a  perceptual  point  of  view.  It 
alone  of  all  solids  appears  constant,  in  form,  from 
whatever  external  point  we  view  it ;  nevertheless  the 
law  of  simplicity  and  economy  holds  in  the  selection 
of  the  cube  as  the  fundamental  standard  of  volume 
because  of  its  obvious  relations  to  the  one  and  two 
dimensional  standards  which  we  have  just  consid- 
ered. 

Any  theory  of  parallel  lines  must  obviously  in- 
volve the  idea  of  surfaces,  hence  before  passing  on 
to  a  direct  study  of  Euclid's  conception  of  parallels 
it  is  necessary  to  take  some  note  of  the  origin  of 
this  idea.  The  conception  of  surfaces  as  geomet- 
rical magnitudes  absolutely  without  thickness  was 
certainly  not  reached  at  a  bound.  Nor  is  it  a  notion 
which  is  rendered  necessary  by  the  essential  nature 
of  a  perceiving  mind.  It  is  essentially  a  metrical 
conception,  and  is  therefore  conditioned  by  those 
characteristic'-' of  things  and  our  experiences  with 
them  which  render  measurements  possible.  Like 
the  conception  of  spatial  equality,  it  could  have  no 
meaning  in  a  homogeneous  fluid  world,  and  in  such 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM       91 

a  world  would  doubtless  never  arise.  The  empirical 
origin  of  this  conception  is  also  shown  by  the  fact 
that  even  the  adult  mind  already  schooled  in  abstract 
thinking-  has  the  greatest  difficulty  in  clearly  con- 
ceiving it.  Perhaps  not  one  in  a  thousand  correctly 
represents  to  himself  the  surface  as  he  has  been 
taught  to  define  it.  It  is  usually  imagined  as  a  c(jr- 
poreal  sheet  of  constant  thickness  which  can  be 
made  small  as  we  please. 

Suppose  we  consider  for  a  moment  the  surface  of 
the  ink  in  the  bottle  before  me.  This  surface,  we 
say,  is  the  boundary  which  separates  the  ink  from 
the  air  that  is  above  it.  But  what  is  this  boundary  ? 
Is  it  ink?  Certainly  not,  for  then  we  should  still 
need  a  boundary  to  separate  that  ink  from  the  air 
above  it.  For  the  same  reason  we  cannot  say  that 
it  is  air.  If,  then,  we  should  magnify  the  two  as 
much  as  we  please  and  should  find  that  they  remain 
always  homogeneous,  each  filling  up  the  space  adja- 
cent to  the  other,  we  should  still  have  to  say  that  the 
surface  between  them  is  neither  the  one  nor  the 
other ;  it  is  not  a  layer  of  air.  of  ink.  of  ether,  nor  of 
space;  it  is  simply  a  boundary  or  geometrical  sur- 
face, and  as  such  occupies  no  space  of  all.  And  yet 
as  we  reach  this  conclusion  how  difficult  it  is  to 
avoid  falling  back  upon  that  class  of  experiences  by 
abstraction  from  which  this  conception  was  orig- 
inally obtained,  and  thus  picturing  to  ourselves  an 
exceedingly  thin  material  partition,  or  at  least  some 


92  PSYCHOLOGIC  ASPECTS  OP  THE  PROBLEM 

small  portion  of  space  which  might  serve  to  keep 
the  two  substances  separate  and  distinct  from  each 
other. 

This,  of  course,  does  not  satisfy  reason.  To  put 
a  space  boundary  between  two  adjacent  portions  of  a 
continuous  quantity  is  not  only  to  regard  space  as 
a  spread-out,  empty  reality  which,  while  providing 
"  room  "  for  things  and  permitting  them  to  approach 
or  recede  from  each  other,  nevertheless  keeps  them 
asunder;  but  it  is  also  to  propose  the  old  problem 
over  again  while  doubling  its  difficulty,  for  we  now 
want  a  boundary  between  the  space  in  question  and 
each  portion  of  the  quantity  separated  by  it.  It  is 
the  same  difficulty  which  is  met  with  in  all  limiting 
conceptions.  It  arises  from  the  disparity  always  to 
be  felt  between  actual  experience  and  those  intel- 
lectual ideals  the  peculiar  character  of  which  this 
experience  itself  suggests  and  determines. 

Coming  now  directly  to  Euclid's  theory  of  parallel 
lines,  we  note  that  it  was  also  determined  to  be  what 
it  is  by  a  variety  of  empirical  data.  If  these  data 
had  been  decidedly  different  from  what  they  are 
some  Lobatchewsky  might  ultimately  have  given  us 
the  Euclidean  system,  but  this  system  would  cer- 
tainly not  have  been  first  in  the  order  of  develop- 
ment. The  data  in  question  are  many  and  simple, 
and  were  certainly  familiar  even  to  the  most  ancient 
civilizations.  The  ornamental  designs  of  savage 
tribes  in  weaving,  drawing,  wood-carving,  and  the 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM. 


93 


like  often  suggest  that  the  triangle's  angle  sum  is 
two  right  angles. 

By  folding  a  triangular  piece  of  cloth  or  paper  as 
shown  in  the  figure,  this  truth  is  directly  perceived. 
The  angles  of  the  triangle  are  seen  to  form  a  straight 
angle  by  bringing  their  vertices  together  at  C'.*-* 


The  same  truth  doubtless  suggested  itself  over 
and  over  again  to  the  workmen  in  clay  and  stone  of 
Babylonia,  Egypt,  and  Greece  in  the  mosaics  and 
pavements  which  they  are  known  to  have  made  from 
differently  colored  stones  of  the  same  shape.  In 
this  way,  too,  it  was  easily  found  that  the  plane 
"  space  "  about  a  point  can  be  completely  filled  only 
by  three  kinds  of  regular  polygons,  that  is,  by  six 
equilateral  triangles,  by  four  squares  and  by  three 
regular  hexagons,"  '^  and  hence  that  this  space  is 
always  equal  to  four  right  angles. 

In  this  ancient  paving,  triangular  stones  of  the 
same  shape  and  size  were  frequently  used  by  placing 

i*Tylor  suggests  this  as  the  probable  origin  of  this  theo- 
rem.    Anthropology,  New  York,  1896,  p.  320. 

15  Proclus  attributes  this  theorem  to  the  Pythagoreans. 
Gow,  op.  cit.  foot  note  p.  143. 


94 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 


their  bases  on  the  same  straight  line  as  in  the  accom- 
panying figure.^  ^ 


Here  the  system  of  equally  distant  lines  is  a  strik- 
ing feature.  It  is  also  made  clear  by  this  figure 
that  the  sum  of  the  interior  angles  on  the  same  side 
made  by  the  intersection  of  any  two  of  these  equally 
distant  lines  by  a  third  line  is  two  right  angles. 
The  obvious  fact  that  this  style  of  paving  may  be 
extended  without  limit  leads  naturally  to  the  convic- 
tion that  parallel  lines  will  not  meet  however  far 
produced,  for  such  lines  are  seen  to  be  ideally  unlim- 
ited and  yet  everywhere  equally  distant.  Thus  we 
have  suggested,  at  least,  in  this  simple  figure  all  the 
empirical  data  necessary  to  the  formation  of  the 
Euclidean  theory  of  parallel  lines. 

But  by  whatever  fact  or  class  of  facts  this  theory 
was  originally  suggested,  the  mechanical  arts  furnish 

16  See  E.  Mach's  valuable  article  in  The  Monist,  Vol.  XTL, 
pp.  481-515. 


PSYCHOLOGIC  ASPECTS  OF  THU.  PROBLEM.       ys 

today  innumerable  examples  of  its  practical  truth. 
The  existence  of  similar  figures  of  unequal  sizes  ami 
the  actual  construction  of  rectangles  whose  angles  all 
remain  right  angles  and  \vh(jse  opposite  sides  con- 
tinue to  be  equal  when  tested  l)y  the  most  accurate 
measuring  instruments  are  constantly  recurring 
proofs  of  Euclid's  validity.  If.  however,  in  the 
construction  of  quadrilaterals  with  angles  all  right 
angles,  and  sides  practically  the  straightest  possible. 
it  had  uniformly  been  found  that  the  opposite  sides 
are  unequal,  as  actually  happens  in  surveying  such 
figures  of  large  size  upon  the  surface  of  the  earth, 
we  should  then  doubtless  have  reached  with  equal 
confidence  a  different  conclusion,  and  our  sciences 
of  mechanics,  physics  and  astronomy  would  have 
been  quite  different  from  what  they  are;  but  as  a 
matter  of  history  this  has  not  been  true.  From  the 
crudest  measurements  to  the  most  refined ;  from  the 
ancient  "  Harpedonaptai  "  *^  of  Egypt  to  the  present 
day  in  the  constantly  increasing  refinement  of  the 
powers  of  accurate  observation,  man's  ability  to 
measure  slight  deviations  from  right  angles  and 
from  the  equality  of  distances  has  remained  rela- 
tively  equal.     There   are  today   no  observed   phe- 

iT  Egyptian  "  Rope  Stretchers  "  mentioned  by  Democritus. 
Their  principal  duty  was  the  construction  of  right  angles  by 
means  of  ropes  divided  into  3,  4-  and  S  equa'  ""•''^  ^^  IciiRth 
See  Cantor's  Vorlesungen  iibcr  Ccschiclilr  dcr  Mathcmotik. 
Vol.  I.,  p.  64,  Leipzig,  1894. 


96       PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

nomena  coming  within  tlie  scope  of  the  physical 
sciences  that  seem  to  contradict  in  any  appreciable 
degree  the  Euclidean  postulate.  Therefore  even 
when  we  set  aside,  as  we  must,  Kant's  contention 
for  the  a  priori  synthetic  nature  of  the  parallel  postu- 
late, it  may  still  be  claimed  that  there  is  no  law  of 
nature  reached  by  scientific  induction  that  can  be 
said  to  have  so  good  a  right  to  be  called  funda- 
mentally and  universally  true  of  the  world  as  we 
know  it. 

As  just  stated,  the  .parallel  postulate  is  rooted 
psychologically,  in  so  far  as  direct  perception  is  con- 
cerned, in  the  ability  to  discriminate  slight  varia- 
tions in  the  size  of  angles,  in  the  length  of  lines,  and 
in  the  departure  of  the  latter  from  "  absolute 
straightness."  It  requires  that  this  ability  shall  re- 
main as  a  rule,  relatively  speaking,  precisely  equal  in 
these  difTerent  directions.  Now  it  is  obvious  that 
'*  the  relations  between  the  least  perceptible  differ- 
ence of  the  angles  of  a  parallelogram  and  the  least 
perceptible  differences  of  the  lines  forming  its  sides 
are  exceedingly  complex  and  variable."  ^^  It  is 
therefore  possible  that  the  law  of  these  relations 
when  determined  by  the  most  refined  experimental 
analysis  ^^  w\\\  prove  to  be  in  essential  agreement 

i»  Ladd :     A  Theory  of  Reality,  New  York,  1899,  p.  315. 

1^  The  writer  has  already  begun,  at  Professor  Ladd's  sug- 
gestion, the  experimental  problem  here  referred  to.  So  far, 
however,  the  results  obtained  are  not  of  a  very  positive  char- 
acter. 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 


97 


with  the  EucHdean  postulate.  Indeed  this  may  be 
even  confidently  expected  in  view  of  the  general 
experiences  of  the  race  to  which  we  have  referred. 

This  conclusion,  however,  is  by  no  means  to  be 
regarded  as  a  necessary  one.  In  fact,  that  ideal 
exactitude  which  is  required  to  establish  beyond 
doubt  the  validity  of  this  postulate  is  no  more  to  be 
expected  in  this  than  in  any  other  form  of  empirical 
testing.  The  absolute  validity  of  Euclid  can  never 
be  established  by  any  mere  appeal  to  perceptual  ex- 
perience, however  refined.  Our  space  may  possibly 
be  proved  in  this  way  to  be  non-Euclidean,  but  it  can 
never  be  shown  to  be  exactly  Euclidean. 

The  analysis,  then,  of  the  experiences  involvetl  in 
the  theory  of  parallel  lines  requires  the  considera- 
tion of  a  special  relation  between  angles  and  dis- 
tances, and  the  problem  is  to  determine  the  peculiar 
character  of  this  relation. 

We  have  already  seen  (Chap.  I.,  pp.  9  and  10) 
that  in  defining  angles  as  differences  of  direction, 
straight  lines,  as  those  which  do  not  change  their 
direction,  and  parallels,  as  straight  lines  having  the 
same  direction,  we  overleap  at  a  bound  the  difficul- 
ties involved  in  the  parallel  postulate.  It  is  there- 
fore evident  that  the  peculiar  properties  of  parallel 
lines  are  somehow  bound  up  in  the  word  direction, 
and  could  be  distinguished  if  the  meaning  of  this 
word  were  subjected  to  a  careful  analysis.  Through 
the  recognition  of  the  importance,  for  Euclid,  of  the 


98  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

idea  of  direction.  Von  Cyon  -"  was  led  to  believe 
that,  in  discovering,  as  he  thought,  a  special  sense- 
organ  in  the  semicircular  canals  for  the  perception  of 
three  fundamental  directions  corresponding  to  the 
co-ordinates  of  the  Cartesian  system,  he  had  solved 
the  whole  space  problem  completely.  But  the  fact 
already  pointed  out  (Chap.  I.,  p.  lo)  that  direction 
as  ordinarily  understood  can  only  be  completely  de- 
fined when  the  parallel  postulate  is  already  assumed 
shows  quite  clearly  that  such  a  solution  is  only  ap- 
parent. It  proceeds,  in  fact,  upon  the  customary 
assumption  expressed  in  the  above  definition  that 
direction  is  the  one  essential,  sufficient,  and  peculiar 
property  of  the  straight  line;  in  other  vv^ords,  that 
straightness  and  direction  are  exactly  equivalent  in 
meaning,  and  that  the  latter  word  is  used  in  pre- 
cisely the  same  sense  in  all  three  of  the  above  defini- 
tions. Now  we  wish  to  show  that  there  is  uncon- 
sciously introduced  into  this  word  as  it  is  used  in 
the  last  two  definitions,  in  addition  to  the  idea  of 
straightness  which  it  always  carries,  an  element  of 
meaning  which  ultimately  arises  from  the  peculiar 
structure  and  symmetry  of  our  bodily  organism. 

It  will  be  remembered  that  in  the  early  part  of  the 
present  chapter  we  called  attention  to  the  peculiar 
symmetrical  arrangement  of  corresponding  sensory 
organs  and  the  influence  of  this  arrangement  upon 

20  PHuegc/s  Archk'  fucr  Physiologic,  1901. 


FSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 


99 


the  orientation  -^  of  the  body  with  reference  to  any 
stimulus  appeaHng  to  the  senses.  This  is  no  doubt 
largely  due  to  the  remarkable  similarity  of  the 
sensory  experiences  of  corresponding  organs,  as.  for 
example,  the  eyes  or  the  ears.  These  sensations, 
however,  are  not  wholly  identical,  and  their  differ- 
ences, combined  with  other  sensation  factors,  mainly 
those  of  movement  brought  out  in  repeated  acts  oi 
adjustment,  give  rise  to  characteristic  distinctions  in 
sensations  of  direction.  For  example,  the  direc- 
tions, before  and  behind,  up  and  down,  right  and 
left,  as  actually  experienced,  involve  sensory  differ- 
ences somewhat  analogous  to  those  of  color.-^ 
These  differences  attach  themselves  to  our  notion  of 
direction,  and  when  unconsciously  carried  over  with 
this  notion  into  the  abstract  space  of  mathematics 
they  lead  to  confusion.  There  is  no  such  thing  as  a 
difference  of  direction  in  geometrical  space.  In  such 
space  tzco  points  are  always  required  to  determine 


-1  This  orientation  may  even  be  wholly  involuntary,  as  in  the 
case  of  the  moth  when  it  helplessly  flies  into  the  flame  and 
is  burned. 

--  It  is  an  interesting  fact  that  Helmholtz  was  led  to  inves- 
tigate the  problem  of  space  by  the  analogy  which  he  perceived 
between  space  and  the  color  system  as  tridimensional  manifolds. 
We  shall  see  when  we  come  to  consider  the  impossibility  of 
carrying  to  one  space  the  same  metrical  standard  which  was 
employed  in  another  that  the  so-called  space-constants  arc 
qualitatively  different  and  yet  serially  related  to  each  other  in 
a  manner  which  is  in  all  essential  respects  similar  to  what  wc 
experience  in  the  perception  of  differences  of  color. 


100     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

a  line.  If  a  number  of  lines  are  conceived  of  as 
going  out  from  a  point  they  can  only  be  distin- 
guished by  an  actually  perceived  or  else  an  imagined 
relation  to  our  bodily  selves.  We  can  transport 
ourselves  in  imagination  to  the  vertex  of  any  angle 
formed  in  this  way,  and  by  the  use  of  a  distinction 
not  inherent  in  the  figure  itself  we  can  represent  the 
angle  to  ourselves  as  a  difference  of  direction,  and 
thus  distinguish  the  lines  from  each  other.  In  no 
other  way  can  we  do  this ;  we  picture  ourselves  as  it 
were  standing  at  the  vertex  of  the  angle,  looking 
alternately  down  its  sides,  and  thus  in  imagination 
sensing  their  directions  as  different. 

We  carry,  then,  into  our  abstract  space-world  with 
this  notion  of  direction  a  misleading  subjective  dis- 
tinction. The  bodily  self  enters  into  spatial  rela- 
tions with  the  other  objects  of  this  world  in  a  pe- 
culiar way ;  it  is  not  allowed  to  take  its  place,  as  this 
abstract  conception  of  space  requires  that  it  should, 
merely  as  one  among  many  objects  of  whose  distin- 
guishing peculiarities  except  in  so  far  as  those  quali- 
tative similarities  are  concerned  which  measurement 
requires,  geometry  can  take  no  account  whatever. 

It  is  also  easy  to  see  that  the  same  "  physiolog- 
ical "  element  of  meaning  attaches  to  the  word  direc- 
tion as  employed  in  the  definition  of  parallel  lines. 
For  what  else  can  be  meant  by  saying  that  such  lines 
have  the  same  direction  and  that  even  to  infinity? 
This  is  clearly  apparent  if  we  follow  closely  the  Ian- 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM      loi 

guage  of  Mr.  J.  S.  Mill  in  the  following  passage :  -^ 
"  Though,  in  order  actually  to  see  that  two  given 
lines  never  meet,  it  would  be  necessary  to  follow 
them  to  infinity ;  yet  without  doing  so  we  may  know 
that  if  they  ever  do  meet,  or  if,  after  diverging  from 
one  another,  they  begin  again  to  approach,  this  must 
take  place  not  at  an  infinite  but  at  a  finite  distance. 
Supposing,  therefore,  such  to  be  the  case,  we  can 
transport  ourselves  thither  in  imagination,  and  can 
frame  a  mental  image  of  the  appearance  which  one 
or  both  of  the  lines  must  present  at  that  point,  which 
we  may  rely  on  as  being  precisely  similar  to  the 
reality."  ^'^ 

But  the  mere  straightncss  of  tivo  lines  lying  in  the 
some  plane  cannot,  of  itself,  justify  any  statement 
as  to  whether  they  will  or  7cill  not  meet  zchen  pro- 
longed without  limit.     If  it  could,  there  would  then 


23  Logic.  Book  ii.,  Chap.  V.,  §  5. 

-^  It  is  interesting  to  note  while  this  passage  is  before  ii'' 
that  its  closing  statement  is  open  to  criticism.  On  a  preced- 
ing page  Mr.  Mill  has  stated  that  "  we  should  not  be  author- 
ized to  substitute  observation  of  the  image  for  observation  of 
the  reality,  if  we  had  not  learnt  by  long  continued  experience 
that  the  properties  of  the  reality  are  faithfully  represented  in 
the  image."  Now  it  is  evident  that  experience  can  only  tell 
us  this  in  the  case  of  realities  and  images,  both  of  which  have 
been  experienced  :  both  must  be  known  before  we  have  a  right 
to  say  that  the  one  faithfully  represents  the  other.  Hence  in 
admitting,  as  Mill  here  does,  the  universality  of  the  truth  of 
the  parallel  postulate,  he  introduces  a  factor  of  knowledge 
which  can  not,  strictly  speaking,  be  gotten  from  experience 
alone  as  he  conceives  it. 


102     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

be  no  need  whatever  of  the  parallel  postulate,  for 
Euclid  becomes  established  at  once  when  the  exist- 
ence of  straight  lines  in  this  sense  is  granted.  '  It  is 
difficult  to  grasp  the  truth  of  this  statement,  so 
natural  has  it  become  through  long-continued  asso- 
ciation of  these  ideas  to  look  upon  "  straightness  " 
and  "  direction  "  in  their  adjective  relations  as  per- 
fectly synonymous  terms. 

Thus  it  has  come  to  appear  that  if  only  the  lines 
are  actually  straight  the  parallel  postulate  must  fol- 
low of  necessity,  and  that  it  is  only  when  we  have 
unconsciously  or  purposely  introduced  some  sort  of 
bending  or  curvature  into  the  lines  themselves  that 
this  postulate  fails.  But  careful  reflection  upon  the 
essential  meaning  of  straight  lines  when  carried 
back  to  the  source  in  experience  whence  this  concep- 
tion has  sprung  reveals  that  this  a  priori  necessity 
of  the  parallel  postulate  simply  does  not  exist. 
|:  Some  distinction  between  the  idea  of  straightness 
and  that  of  direction  must  obviously  be  maintained 
:  if  any  non-Euclidean  system  is  to  be  taken  seriously 
jas  having  anything  to  do  with  reality.  If  these  two 
/  words  are,  in  fact,  identical  in  meaning  and  the  word 
"  direction  "  as  used  in  the  definitions  of  straight 
I  lines,  angles,  and  parallels  as  given  above  holds  pre- 
cisely the  same  significance  in  each  definition,  it  is 
easy  to  see  that  any  serious  struggle  between  Euclid 
and  his  modern  rivals  is  out  of  the  question,  for 
Euclid  alone  is  left  on  the  field. 


FSVCHOLOGIC  ASPUC'IS  OF  THE  PROBLEM      lo.^ 

A  familiar  illustration  will  assist  in  making  this 
distinction  clear.  As  everyone  knows,  a  curve  is 
frequently  defined  in  elementary  geometry  as  a  line 
which  changes  its  direction  at  every  point.  Now  if 
we  substitute  straightness  for  direction  in  this  defini- 
tion it  becomes  at  once  absurd.  For  even  if  we 
could  give  the  resulting  expression,  "  changes  its 
straightness,"  etc.,  an  intelligible  meaning  by  re- 
garding it  as  signifying  the  amount  of  angular  devia- 
tion from  the  tangent  to  the  curve  at  the  point 
considered,  the  definition  would  still  break  down 
when  applied  to  the  circle,  for  in  this  case  we  have 
a  figure  whose  angular  deviation  from  the  tangent 
line  is  a  constant  quantity,  and  therefore,  according 
to  this  definition,  the  circle  could  not  be  a  curve 
at  all. 

We  see,  then,  that  the  word  "  direction  "  names  a 
-complex  idea  which  is  altogether  too  inclusive  and 
variable  in  meaning  to  specify  accurately  what  is 
meant  by  the  straight  line.  It  leads  to  confusion 
where  precision  of  statement  and  accuracy  of  mean- 
ing are  urgently  demanded,  and  should  therefore  be 
avoided  if  possible. 

Generally  speaking,  the  confusion  so  frequently 
met  with  in  discussions  on  the  philosophic  founda- 
tions of  geometry  arises  from  the  general  tendency 
to  regard  as  simple  and  without  the  need  or  the  i)OS- 
sibility  of  further  analysis  certain  ideas  which  m 
reality  are  complex  in  character  and  therefore  can- 


104     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

not  be  regarded  as  ultimate.  Emphasis  is  laid  now 
upon  one  phase,  now  another,  of  these  complex 
ideas,  and  the  whole  problem  is  solved  by  overlook- 
ing entirely  the  real  question  at  issue, 
-^  Of  all  the  fundamentals  of  geometry,  that  which 
stands  most  in  need  of  satisfactory  analysis  is  the 
conception  of  the  straight  line.  What  is  there  in 
this  word,  let  us  ask,  that  makes  it  not  only  univer- 
sally applicable,  but  even  indispensable  to  all  forms 
of  geometry?  What  do  we  and  what  ought  we  to 
mean  by  "  straightness  "  as  applied  to  lines  in  all 
these  systems? 

This  conception,  by  virtue  of  its  complication  with 
certain  metrical  ideas  that  have  always  been  asso- 
ciated with  it,  gives  philosophically  the  greatest 
trouble  in  any  attempted  critical  estimation  of  differ- 
ent geometrical  systems.  Angles  present  no  diffi- 
culties of  so  serious  a  character.  The  old  Euclid- 
ean postulate  that  all  right  angles  are  equal  is  in 
reality  a  theorem  which  has  lately  been  rigorously 
proved,^^  and  is  just  as  true  of  non-Euclidean  as  it  is 
of  Euclidean  geometry.  The  angular  magnitude 
about  a  point  is  equal  to  four  right  angles  in  any  un- 
bounded geometrical  surface  whose  curvature  is  con- 
stant,   and   is   therefore  the  same  in  all   forms  of 


25  Killing,  Gnmdlagen  der  Geometrie  (Paderborn,  1898), 
Vol.  II.,  p.  171 ;  and  especially  Hilbert,  Gnmdlagen  der  Geom- 
etrie, Leipzig,  1899,  p.  16. 


PSYCHOLOGIC  ASPECTS  OP  THE  PROBLEM      105 

geometry.^*'  Hence  the  creation  of  new  geometries 
does  not  essentially  modify  the  difficulties  to  be  met 
with  in  the  treatment  of  angles.  Angular  magni- 
tudes can  always  be  expressed  as  ratios  of  linear 
magnitudes,  and  are  therefore  easily  determined 
when  the  latter  are  known.  Upon  the  possibility  of 
such  ratios  the  whole  science  of  trigonometry  cor- 
responding to  any  conception  of  space  is  erected. 

Let  us,  then,  focus  our  thought  upon  the  straight 
line  and  try  by  careful  analysis  to  answer  our  ques- 
tions. Having  already  traced  the  growth  of  this 
conception  as  it  presents  itself  to  the  consciousness 
of  the  ordinary  man,  let  us  now  turn  to  the  mathe- 
matician and  endeavor  to  learn  from  him  what  he 
considers  to  be  essential  to  the  straight  line  as  shown 
by  his  definitions.  When  the  results  of  this  analysis 
have  been  attained  we  shall  then  try  to  relate  them. 
if  possible,  to  actual  experience. 

We  have  seen  that  Riemann  makes  room  for  an 
unlimited  number  of  geometries  differing  from  each 
other  fundamentally  in  the  meaning  assigned  to  the 
linear  element  ds,  and  that  in  the  further  working 
out  of  his  system  emphasis  was  laid  upon  cun'a- 
ture^'^  as  an  important  conception.     Cayley  makes 


28  There  may  appear  to  be  an  exception  to  this  statement  in- 
the  case  of  the  apex  of  the  cone  where  this  anprular  magnitude 
is  always  less  than  four  right  angles,  but  I  have  attempted  to 
remove  this  objection  by  the  word  unbounded. 

27  We  have  already  noted  the  development  of  this  conccp- 


io6    PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

distance  fundamental,  and  with  the  aid  of  Klein  I 
shows  that  a  variation  of  the  laws  by  which  distance 
is  measured  is  all  that  is  necessary  to  distinguish 
Euclidean  and  non-Euclidean  systems.  From  this 
it  appears  that  geometries  are  to  be  distinguished  ■, 
by  the  number  and  character  of  the  postulates  em-  1 
ployed  to  determine  certain  conceptions  which  for 
the  ordinary  man  define  the  straight  line.  For  him, 
as  we  have  said,  this  line  is  at  all  times  simply  the 
shortest  distance  between  two  points  or  else  a  line 
which  does  not  change  its  direction  at  any  of  its 
points.  But  these  words  are  not  simple  in  meaning, 
and  therefore  stand  in  need  of  definition  themselves. 
Direction  has  already  been  considered  and  need  no 
longer  detain  us.  Let  us  now  examine  the  phrase 
"  shortest  distance  "  and  endeavor  to  determine  its 
meaning.  Reflection  shows  that  this,  too,  is  far 
from  being  a  simple  conception.  It  presupposes,  in 
fact,  all  the  assumptions  necessary  for  measurement. 
It  presumes  beforehand  the  possibility  of  measuring 
all  kinds  of  lines  that  can  be  drawn  anywhere  in 
space,  else  how  could  it  be  said  that  a  certain  one  of 
these  lines  is  the  shortest  distance  between  two  given 
points.  This  is  certainly  a  tremendous  assumption. 
All  the  postulates  demanded  by  metrical  geometry 
are  wrapped  up  in  this  definition.     Before  measure- 

tion  in  the  historical  treatment  of  Riemann's  geometry  in 
Chapter  I.*,  we  shall  take  it  up  for  more  thorough  treatment  in 
our  last  chapter  in  the  discussion  of  space  conceptions. 


FSYCHOLUGIC  ASPECTS  OF  THE  PROBLEM       107 

tnent  is  possible  we  must  have  a  standard  of  meas- 
urement, and  this  standard  is  itself  the  straight  line. 
Measurement,  then,  presupposes  the  straight  line  as 
a  necessary  pre-condition  of  its  own  possibility,  and 
therefore  cannot  be  taken  as  a  simpler  and  more 
ultimate  notion  with  which  to  define  the  straight 
line.  Furthermore,  the  existence  of  a  minimum  is 
itself  an  assumption  which  involves  certain  logical 
consequences  of  an  interesting  character,  which  can- 
not be  dealt  with  here.  Finally,  this  definition  fails 
in  the  case  of  straight  lines  which  join  antipodal 
points  of  double  elliptic  space.  Between  such  points 
there  is  an  infinity  of  shortest  lines. 

These  are  difficulties  which  are  certainly  of  a  seri- 
ous character.  Nevertheless  it  seems  hardly  possi- 
ble wholly  to  avoid  some  conception  of  distance 
when  we  talk  of  straight  lines.  Long  ago  Leibnitz 
made  distance  fundamental,^^  and  the  same  point  of 
starting  has  recently  been  taken  by  Frischauf  and 
Peano.  Peano  defines  the  straight  line  ab  as  a  class 
of  points  ,v,  such  that  any  point  y,  whose  distances 
from  a  and  b  are  respectively  equal  to  the  distances 
of  .r  from  a  and  b,  must  be  coincident  with  x.  But 
Peano  fails  to  prove  either  that  such  a  line  exists 
or  that,  if  it  does  exist,  it  is  determined  by  any  two 
of  its  points.2»  This,  of  course,  is  impossible  with- 
out the  use  of  certain  special  axioms.    According  to 

2"  Russell,  Principles  of  Mathewatics,  Vol.  I.,  p.  410- 
2»  Russell,  Principles  of  Mathematics.  Vol    T..  p.  410  flF. 


io8     PSYCHOLOGIC  ASPEC'TS  OF  THE  PROBLEM 

Peano,  five  such  axioms  are  needed.  The  group 
which  he  has  selected  is  an  interesting  one,  because 
it  defines  distance  by  the  use  of  hetn'cen  as  an  in- 
definable notion.  His  axioms  for  the  straight  line 
ab  are  as  follows  :  ( i )  Points  hetzveen  which  and  b 
the  point  a  lies;  (2)  the  point  a;  (3)  points  betzvcen 
a  and  6;  (4)  the  point  b;  (5)  points  between  which 
and  a  the  point  b  lies.'"'  Just  what  is  meant  by  be- 
tzveen  is  nowhere  clearly  explained. 

Vailati  attempts  an  explanation  which  is  rejected 
by  Peano  ^^  on  the  ground  that  betzveen  is  a  relation 
of  three  points  and  not  of  two  only.  As  a  matter 
of  fact  if  we  confine  ourselves  to  projective  geometry 
even  three  points  on  a  line  are  not  so  related  that 
any  one  of  them  can  be  said  to  be  betzvcen  the  other 
two.  The  word,  between,  involves  a  certain  order- 
ing of  the  points  which  in  projective  geometry  de- 
pends upon  the  nature  of  the  quadrilateral  construc- 
tion which  requires  for  its  proof  a  point  outside  its 
own  plane  and  hence  is  not  possible  without  three 
dimensions.  It  also  requires  four  perspective  tri- 
angles. The  generation  of  order  by  this  method  is 
therefore  considerably  complicated  and  the  simplest 
proposition  involving  betzveen  which  remains  unal- 
tered by  projection  is  one  which  requires  four  points. 

It  is  obvious  then  that  if  ''  between  "  is  to  be  estab- 
lished at  all  as  a  unique  relation  of  any  tzifo  points 

3"  Rivista  di  Matematica,  Vol.  IV.,  p.  62. 
^'^Rh'isfa  di  Matematica,  Vol.  I.,  p.  393. 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM      109 

it  cannot  be  done  by  any  appeal  to  projective  con- 
siderations. But  in  spite  of  this  Russell  ^-  sets 
Peano's  criticism  aside  as  inadequate  and  adopts 
what  is  practically  Vailati's  position.  He  avoids  the 
word  "  between,"  however,  and  introduces  in  its 
stead  a  new  indefinable.  He  posits  a  class  of 
asymmetrical  transitive  relations  ^^  which  he  calls  K 
and  assumes  that  betw'een  any  two  points  there  is 
one  and  only  one  relation  of  this  class.  Eight 
axioms  which  Russell  shows  to  be  distinct  and  mu- 
tually independent  are  required,  as  he  thinks,  to 
define  what  is  meant  by  this  class  of  relations.'*^ 

Avoiding  his  symbolism  these  axioms  may  be 
stated  as  follow^s:  (i)  There  is  a  class  of  asym- 
metrical transitive  relations  K;  (2)  there  is  at  least 
one  point,  and  if  R  be  any  term  of  K  we  have;  (3) 
R  is  an  aliorelative,  that  is,  a  relation  which  no  term 
has  to  itself;  (4)  the  converse  of  R  is  a  term  of  K; 
(5)  R2  =  R,  (6)  the  domain  of  the  converse  of  R 


32  Principles  of  Mathematics,  Vol.  I.,  pp.  394  ff- 
^^  In  the  sense  in  which  these  terms  are  employed  by  Mr. 
Russell  they  may  be  explained  as  follows :  When  the  converse 
of  a  relation  is  the  same  as  the  relation  itself,  the  relation  is 
said  to  be  symmetrical ;  but  when  the  converse  and  the  orip- 
inal  relation  are  incompatible  the  relation  is  said  to  be  asym- 
metrical. Examples  of  the  former  are  such  as  identity,  equal- 
ity, and  inequality;  and  of  the  latter  such  as  better  and  worse, 
greater  and  less.  A  relation  is  transitive  when  any  such  con- 
dition as  the  following  holds :  viz..  if  a  be  similar  to  b  and  b 
similar  to  c.  then  a  is  similar  to  c. 
^*  Princip.  Math.,  Vol.  I.,  pp.  .WS-.^Q^- 


no  PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

is  contained  in  the  domain  of  R;  (7)  between  any 
two  points  there  is  one  and  only  one  relation  of  the 
class  K;  (8)  if  a,  h  be  points  of  the  domain  of  R, 
then  either  a  holds  the  relation  R  to  &  or  t  holds 
this  relation  to  a.  The  seventh  axiom  is  obviously 
double.  It  asserts  ( i )  that  there  is  one  such  relation 
between  any  two  points,  and  (2)  that  there  is  only 
one.  The  first  member  of  the  group  is  not  an  axiom 
at  all  but  only  the  assumption  of  the  indefinable  class 
of  relations  K. 

With  this  outfit  of  assumptions  the  various  forms 
of  geometry  may  be  constructed.  The  mutual  inde- 
pendence of  the  entire  group  is  shown  in  the  usual 
way.  Any  one  of  them  may  be  denied  and  a  logic- 
ally consistent  system  built  upen  what  remains. 
The  fifth  and  seventh  axioms  are  the  most  interest- 
ing here.  In  ( 5 )  we  may  deny  either  that  R  is  con- 
tained in  R2  or  that  R^  is  contained  in  R.  If  we 
deny  the  former  the  resulting  straight  line  becomes 
a  series  of  points  which  does  not  possess  the 
"  density  "  or  "compactness  "  which  the  construc- 
tions of  Euclid  and  non-Euclid  require.  The  gen- 
eral result,  however,  is  logically  sound,  the  series  in 
question  simply  lacks  the  degree  of  continuity  which 
geometry  requires.  If  we  deny  the  latter,  the  result 
proves  to  be  untrue  of  angles  which  otherwise  may 
be  made  to  satisfy  all  the  conditions  expressed  by 
this  group  of  axioms.  If  a  Euclidean  and  a  hyper- 
bolic space  be  considered  together  all   the  axioms 


PSYCHOLOGIC  ASPECTS  OP  THE  PROBLEM      in 

Still  hold  except  the  first  part  of  (7).  The  whole 
group  with  the  exception  of  (7)  is  shown  to  be 
necessary,  but  for  (7)  Mr.  Russell  is  only  able  to 
maintain  a  high  degree  of  probability. 

We  have  now  to  consider  another  interesting 
effort  to  reach  the  "  minimum  essential  "  to  the 
notion  of  "  straightness."  This  lays  the  emphasis 
upon  a  different  factor  of  experience  from  those 
just  considered,  and  consequently  appeals  very 
strongly  to  those  who  prefer  to  approach  geometry 
from  the  motor  side,  rather  than  from  that  statical 
conception  of  an  "  empty  "  space  which  may  be 
reached  by  abstraction  from  the  materials  furnished 
by  vision  alone  without  reference  to  bodily  move- 
ments actual  or  imagined. 

The  effort  to  w^hich  we  refer  is  that  of  Pieri.^"^ 
who  proceeds  to  build  geometry  upon  the  two  in- 
definables.  point  and  motion.  He  defines  the  straight 
line  joining  two  given  points  as  the  class  of  points 
whose  internal  relations  remain  unchanged  by  any 
motion  which  leaves  the  two  points  fixed.  His 
system  is  simple  and  logically  unimpeachable.'"  But 
here  again  we  are  dealing  with  a  complex  idea. 
Motion  as  used  by  Fieri  is  not  simply  the  motion  <if 
a  single  point,  but  a  certain  law  of  motion  is  assumed 
along  with  this  simpler  idea.     We  have  the  motion 

^■' Delia  gcometria  elcmcntarc  como  sisfcma  ipotitico  dcdut- 
tivo,  Turin,  1899. 

36  Russell,  Principles  of  Math.,  p.  410. 


112     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

of  a  number  of  points  taking  place  in  such  a  way 
as  to  leave  certain  relations  unchanged.  In  other 
words  it  is  the  motion  of  a  rigid  body  which  is  here 
assumed. 

If  then  we  ask  how  motion  as  thus  understood  is 
to  be  distinguished  from  other  transformations  it 
becomes  readily  apparent  why  Fieri  was  able  to  con- 
struct metrical  geometry  by  considering  such  a  no- 
tion as  indefinable.  For  when  we  attempt  to  answer 
this  question  we  see  that  in  this  idea  of  motion  all 
the  conditions  necessary  to  metrical  geometry  are 
already  assumed.  Fieri  means  simply  a  transforma- 
tion which  leaves  all  essential  metrical  properties 
and  conditions  unchanged.  Having  thus  presup- 
posed metrical  properties  in  his  conception  of  mo- 
tion, he  then  turns  about  and  defines  these  same 
properties  in  terms  of  this  conception.  Little  wonder 
that  he  succeeds. 

It  must  be  observed  also  that  Riemann's  famous 
dissertation  involves  the  same  difficulty.  He  as- 
sumed that  the  linear  element  ds  remains  unaltered 
by  the  same  infinitesimal  motion  of  all  of  its  points, 
and  bases  his  system  upon  this  assumption.  He 
thus  presupposes  the  existence  of  equal  spatial  quan- 
tities in  different  places  which  is  equivalent  to  the 
assumption  of  spatial  homogeneity,  or  the  free  mo- 
bility of  rigid  bodies.  In  other  words  by  assuming 
metrical  properties  in  his  ds  and  then  proceeding  to  i 
determine  these  properties  upon  the  basis   of  this 


PSYCHOLOGIC  ASPECTS  OI-   THE  PROBLEM      iij 

assumption,  he  easily  draws  out  at  the  faucet  what 
he  has  already  poured  in  at  the  bung. 

Glancing  back  now  over  the  definitions  which  we 
have  just  considered  in  the  light  of  the  critical  ex- 
amination to  which  we  have  subjected  them  let  us 
briefly  summarize  the  resulting  facts.  We  have 
found  that  in  each  case  certain  metrical  properties 
at  some  stage  or  other  in  the  process  were  either 
unconsciously  assumed  or  else  openly  postulated. 
Peano  starts  with  distance  and  closes  his  discussion 
by  grounding  this  conception  upon  "  between  "  as 
an  indefinable  notion.  Russell,  though  holding  es- 
sentially the  same  meaning,  dispenses  with  this  word 
and  substitutes  in  its  place  a  particular  class  of 
asymmetrical  transitive  relations  by  means  of  which 
the  order  of  points  on  tlie  straight  line  may  be  gen- 
erated in  a  perfectly  definite  way.  And  finally  we 
found  Fieri  employing  in  his  definition  the  idea  of 
motion  which  he  assumed  to  take  place  in  such  a 
way  as  to  allow  the  unique  character  of  the  straight 
line  as  well  as  certain  metrical  properties  to  remain 
wholly  unchanged. 

If  now  we  sever  these  words  from  their  abstract 
logical  surroundings  and  endeavor  to  carry  them 
back  to  a  meaning  in  experience,  we  find  that  each 
word  in  its  turn  bears  with  it  what  is  formally  essen- 
tial to  the  idea  of  the  straight  line.  All  the  way 
through,  this  conception  of  the  line  as  reached  by 
abstraction  from  certain  forms  of  experience  already 


114     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

pointed  out,  and  defined  as  the  shortest  distance 
between  two  points,  or  as  a  geometrical  object  which 
when  rotated  about  any  two  of  its  points  remains 
always  within  itself,  has  really  been  present  deter- 
mining the  result.  It  is  this  conception  undoubt- 
edly which  has  guided  Peano  and  Russell  in  select- 
ing the  postulates  by  which  their  conceptions  are  at 
last  defined.  It  is  this  conception  and  naught  else 
which  has  determined  the  number  and  character  of 
these  postulates  to  be  what  they  are.  There  are  cer- 
tainly no  a  priori  reasons  which  independently  of 
those  actual  experiences  in  which  the  notion  of  the 
straight  line  is  ultimately  grounded,  could  have  de- 
cided this  matter.  Why  select  this  peculiar  con- 
ception at  all  as  the  one  most  convenient  and  there- 
fore best  suited  to  serve  as  the  point  of  starting  in 
all  systems  of  geometry,  and  why  ascribe  to  it  just 
those  peculiar  marks  that  we  do?  It  takes  experi- 
ence to  answer  this  question.  No  amount  of  mere 
passive  contemplation  of  blank  space,  if  such  a  thing 
were  possible,  could  ever  lead  to  a  direct  intuition 
of  the  peculiar  properties  of  the  straight  line.  We 
can  reproduce  such  an  object  in  imagination,  it  is 
true,  but  in  reality  this  is  but  an  "  experiment  in 
thought ']  which  could  not  take  place  except  upon 
a  basis  of  past  experience. 

We  do  not  wish  to  be  misunderstood.  What  we 
are  here  contending  for  is  the  empirical  origin  of 
the  special  properties  of  the  straight  line  which  dis- 


PSYCHOLOGIC  ASPECTS  OF  TUli  PROBLEM      ns 

tinguish  it  as  a  peculiar  relation  between  two  points. 
We  do  not  mean  to  assert  that  there  is  nothing  a 
priori  about  this  conception.  The  straight  line  is  a 
peculiar  relation  between  points.  It  both  separates 
and  unites  these  points  in  a  relational  way.  The 
points  can  not  coalesce,  and  become  one  point,  they 
must  be  kept  distinct  from  each  other,  else  the  line 
which  they  are  supposed  to  determine  vanishes  away. 
So  much  is  true  of  every  straight  line  however  its 
peculiar  characteristics  may  otherwise  be  deter- 
mined, w-hether  we  represent  it  to  ourselves  by 
means  of  a  visual,  a  tactual,  or  a  motor  image.  So 
much  at  least  appears  to  be  involved  in  the  very 
notion  of  space  as  an  externalizing  principle  which 
renders  possible  any  system  of  distinct  coexisting 
entities.  Whatever  is  more  than  this  has  a  basis  in 
experience,  and  w  henever  we  conceive  such  a  line  to 
be  something  which  admits  of  definition  it  becomes 
much  more  than  a  mere  abstract  relation  between 
points,  it  becomes  a  straight  relation,  the  question  of 
the  economy  of  effort  represented  as  being  somehow 
felt  in  passing  from  one  point  to  the  other  now 
enters  into  the  conception  and  carries  us  back  at 
once  to  data  which  we  can  only  represent  to  our- 
selves as  being  furnished  by  some  form  of  sensory 
experience,  real  or  imagine'd.  How  shall  I  proceed 
in  the  shortest  time  possible  and  with  a  minimum 
of  effort  from  one  point  to  the  other?  That  path- 
way if  you  please  which  answers  this  f|uestion  is 


ii6    PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

the  straight  Hne.  If  we  now  ask  just  how  this 
pathway  is  to  be  determined,  we  fall  back  upon  cer- 
tain special  experiences  such  as  we  have  already  de- 
scribed. The  line  which  best  satisfies  this  require- 
ment of  a  minimum  of  efifort  for  us  tridimensional 
beings,  is  one  which  is  congruent  with  itself,  in  all 
positions,  which  is  the  shortest  distance  between  two 
points  which  rotates  within  itself,  which  looks  the 
same  from  all  points  not  in  it,  or  one  whose  internal 
relations  remain  unchanged  by  any  motion  which 
leaves  two  of  its  points  fixed.  If  we  neglect  the 
question  as  to  the  number  of  space  cKmensions 
which  are  to  be  taken  account  of,  these  definitions 
all  mean  essentially  the  same  thing,  but  they  empha- 
size somewhat  different  aspects  of  the  complex  ex- 
perience involved. 

This  conception  then  is  a  generalization  from 
facts  which  come  within  the  limits  of  experience 
and  must  therefore  fit  these  facts  when  carried  back 
to  them.  It  is  only  when  we  go  beyond  the  facts 
and,  by  following  the  lead  of  suggestions  offered 
by  experience  or  by  investigating  the  logical  possi- 
bilities involved,  pronounce  upon  such  matters  as 
the  infinity  or  two  sidedness  of  the  straight  line  that 
the  roads  leading  to  Euclid  and  non-Euclid  begin 
to  diverge.  )  This  may  be  made  clear  by  the  follow- 
ing Tllusfration.  Suppose  we  have  two  perfectly 
straight  lines  lying  in  any  position  in  the  same  plane 
and  extending  without  limit  in  both  directions ;  let 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM      117 

one  of  these  lines  rotate  about  a  given  point  not  on 
the  other  Hne,  starting  the  rotation  with  the  lines 
intersecting.  The  point  of  intersection  moves  along 
the  fixed  line  either  away  from  or  toward  the  fixed 
point  of  rotation  according  to  the  direction  in  which 
the  line  is  rotated.  For  convenience  let  it  move 
away  from  this  point,  /rhree  different  results  are 
logically  possible.  When  the  rotating  line  ceases  to 
intersect  the  fixed  line  in  one  direction  it  will  imme- 
diately intersect  in  the  opposite  direction,'*'  or  it  will 
continue  to  rotate  for  a  time  before  intersection 
takes  place,  or  else  there  will  be  a  period  of  time 
during  which  the  two  lines  intersect  in  both  direc- 
tions. The  first  of  these  possibilities  gives  Euclid's, 
the  second  Lobatchewsky's.  and  the  third  Riemann's 
geometry. 

The  mind's  attitude  toward  these  three  possibili- 
ties taken  successively  illustrates  in  a  curious  way 
the  essentially  empirical  nature  of  the  straight  line 
as  we  conceive  it.  Logically  one  of  these  possibili- 
ties is  just  as  acceptable  as  the  other.  From  this 
point  of  view  strictly  taken  there  is  certainly  no 
reason  for  preferring  one  of  them  to  another.  Psy- 
chologically, however.  Riemann's  hypothesis  seems 
absolutely  contradictory,  and  even  Euclid's  is  not 
quite  so  acceptable  as  that  of  Lobatchewsky.  The 
lines  cannot  intersect  in  both  directions,  we  say.  for 

37  We  can  not  say  the  opposite  end  for  the  lines  in  each  case 
and  in  both  directions  are  supposed  to  be  unbounded. 


ii8    PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

in  that  event  they  would  necessarily  be  curved  or  at 
any  rate  not  continuous  straight  lines.  Even  the 
assumption  which  leads  to  Euclid  seems  not  to  be  in 
accord  with  the  lines  of  experience  which  we  call 
parallel;  for  all  such  lines  must  turn  a  little  way 
after  ceasing  to  intersect  at  one  end  before  they 
begin  to  intersect  at  the  other.  The  logical  require- 
ments which  in  this  case  lead  to  Euclid,  seem  to  be 
at  variance  with  those  very  experiences  which  have 
produced  this  geometry.  The  lines  seem  to  be  as 
pictured  by  the  imagination,  necessarily  curved. 
Suppose  that  the  point  about  which  the  rotating 
line  turns  is  at  a  great  distance  from  the  fixed  line, 
it  then  appears  wholly  impossible  to  our  picturate 
thinking  to  represent  the  two  lines  as  straight  under 
the  given  conditions.  They  appear  to  be  geodesies 
joining  antipodal  points  on  the  surface  of  an  im- 
mense sphere.  The  reason  for  this  appearance  is 
obvious.  It  is  because  of  our  familiarity  with  finite 
spherical  surfaces  whose  straightest  lines  are  curved 
and  are  known  to  intersect  in  the  manner  assumed 
by  Riemann.  Hence  in  harmony  with  the  mind's 
general  tendency  to  settle  its  abstract  ideas  upon 
some  convenient  concrete  model  perceived  or  im- 
agined, Riemann's  straight  lines  become  at  once 
attached  to  the  sphere  and  are  thus  pictured  as 
curved.  In  Lobatchewsky's  assumption  the  case  is 
quite  different,  for  it  seems  to  contradict  none  of 
our  ideas  of  the  straight  line.     Beltrami's  pseudo- 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM      ng 

Spherical  surfaces  upon  which  all  the  theorems  of 
Lobatchewsky  may  be  projected  and  there  visualized 
as  figures  whose  sides  are  curved,  are  not  at  all 
familiar  to  us.  But  suppose  these  saddled  shaped 
surfaces  had  been  the  ones  with  which  our  notion  of 
curved  lines  had  always  been  associated  and  that 
pseudo-spheres  and  spheres  had  simply  exchanged 
places  in  our  experience,  would  not  Lobatchewsky's 
assumption  in  that  event  have  appeared  to  be  just  as 
incompatible  w4th  our  notion  of  straight  lines  as 
Riemann's  does  now  ? 

I  Again  there  is  logically  no  reason  why  the  geom- 
etry of  Lobatchewsky  should  have  appeared  in  his- 
tory before  that  of  Riemann.  We  have  seen, 
however,  that  the  former  assumption  worried  Sac- 
cheri  very  profoundly  in  his  heroic  efforts  to 
"  vindicate  Euclid,"  more  than  a  century  and  a 
quarter  prior  to  the  publication  of  Riemann's  dis- 
sertation. It  was  not  a  logical  but  a  psychological 
or  experiential  difficulty  which  caused  Saccheri  to 
reject  the  logical  conclusions  to  which  his  own  labors 
clearly  and  inevitably  pointed ;  and  it  was  certainly 

,  the  same  sort  of  difficulty  which  caused  the  imme- 
diate rejection,  by  himself  and  by  subsequent 
mathematicians,  of  the  assumption  upon  which  Rie- 
mann's geometry  is  grounded. 

We  must  now  briefly  summarize  the  results  of  the 
present  chapter  and  indicate  the  direction  which  m 


120     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

consequence  of  the  positions  here  taken,  our  further 
treatment  of  the  problem  must  follow. 

We  have  endeavored  to  show  that  all  strictly 
metrical  conceptions,  including  the  straight  line  and 
the  theory  of  parallels,  grow  out  of  experience  with 
the  objects  of  our  environment,  and  that  the  particu- 
lar form  which  these  conceptions  have  assumed  is 
I  determined  to  be  what  it  is  by  the  peculiar  character 
1  of  this  experience.  This  is  true  not  only  of  the 
special  conceptions  by  which  the  different  forms  of 
geometry  are  defined,  but  it  is  also  true  of  the 
space  conceptions  which  lie  at  the  foundation  of 
each.  Geometrical  spaces  are  abstract  ideal  concep- 
tions which  arise  out  of  experience  in  much  the 
same  way  and  by  means  of  essentially  the  same 
faculties  as  those  which  are  employed  in  the  develop- 
ment of  other  conceptions.  There  is  nothing  espe- 
cially mysterious  about  them.  They  differ  funda- 
mentally as  to  the  logical  possibilities  which  the 
same  general  experience  suggests  in  regard  to  what 
may  conceivably  happen  beyond  its  actual  realm, 
and  also  as  to  the  particular  aspects  of  this  experi- 
ence upon  which  emphasis  is  especially  laid. 
From  somewhat  heterogeneous  psychological  spaces 
through  the  physical  and  geometrical  on  up  to  that 
higher  conception  which  in  a  sense  must  include 
and  harmonize  them  all.  the  process  is  a  unitary 
one  and  essentially  unbroken.  By  fusing  together 
the  spatial   data   derived    from'  the   various   senses 


PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM      ui 

through  ignoring-  the  differences  between  their  sev- 
eral dehverances  and  by  correcting  the  appearances 
to  one  sense  through  those  of  another,  in  such  a 
manner  as  to  give  the  most  complete  and  trust- 
worthy perception  of  the  objects  of  our  environment 
in  their  relations  to  ourselves  and  to  each  other,  we 
derive  our  notion  of  space  in  which  the  physical 
world  is  apparently  set.  We  simply  manipulate  the 
data  of  sense  as  if  always  with  an  end  in  view, 
namely  that  of  perceiving  the  things  (^f  our  |)hysical 
world  successfully  and  adjusting  ourselves  comfort- 
ably to  them.  At  a  higher  stage  the  same  purposive 
process  yields  those  adjectives  which  define  the 
various  forms  of  geometrical  space.  Just  as  the 
varying  appearances  of  things  to  the  different  senses 
for  example  were  ignored  in  order  to  arrive  at  their 
real  place  in  our  mental  picture  of.  the  world,  so  the 
varying  and  irregular  deformities,  to  which  they  are 
actually  subjected  in  different  places  by  virtue  of 
their  relations  to  each  other,  when  abstracted  from 
lead  directly  to  the  conception  of  relativity  and 
homogeneity  of  space.  In  a  similar  fashion  notions 
of  unboundedness  and  infinite  divisibility  arise.  . 

We  have  pointed  out  the  indispensableness  of  the 
straight  line  and  indicated  its  relation  to  the  parallel  , 
postulate.      The    two    are   bound    together   by    no  j 
logical  necessity  but  by  a  relation  which  is  assumed 
to  hold  in  a  region  which  lies  beyond  all  experiences 
actual  or  possible.     This  conception  r,f  the  straight 


122     PSYCHOLOGIC  ASPECTS  OF  THE  PROBLEM 

line  when  freed  from  certain  ideas  whicl\  by  con- 
tingent, though  long  continued  and  practically  un- 
interrupted associations  have  served  to  disguise  its 
essential  meaning,  is,  so  far  as  it  is  grounded  in 
actual  experience,  the  most  fundamental  figure  in 
all  forms  of  geometry.  Straight  lines  are  straight 
lines  in  non-Euclid  as  well  as  in  Euclid  and 
although  we  may,  to  aid  the  imagination  and  for 
convenience  of  study,  conceive  the  figures  of  these 
newer  geometries  to  be  projected  upon  surfaces  in 
finite  portions  of  Euclidean  spaces  of  higher  dimen- 
sions —  surfaces  whose  lines  are  all  curved ;  yet  we 
must  not  confuse  these  props  of  the  imagination 
with  the  realities  for  which  these  geometries  as 
logical  systems  may  be  supposed  to  stand. 

When  we  go  beyond  the  experiences  which  have 
given  us  this  conception  of  the  straight  line,  as  in 
thought  we  may,  and  consider  the  different  logical 
possibilities  which  lie  open  to  us  there,  we  get  by 
postulating  these  possibilities  in  succession  in  con- 
junction always  of  course  with  those  idealizations 
which  actual  experience  affords,  the  different  sys- 
tems of  geometry. 

There  is  then,  and  must  always  remain  a  gap 
between  empirical  and  mathematical  exactness,  be- 
tween experiences  actual  and  the  ideals  which  ex- 
perience suggests.  In  the  case  before  us  there  is 
such  a  gap  between  the  conditions  imposed  by  sense- 
perception   and   the   mathematical    precision    which 


PSYCHOLOGIC  ASPECTS  OP  PHP  PROBLEM      123 

results  when  these  conditions  are  supposed  to  be 
withdrawn.  In  this  opening,  as  we  have  said, 
logical  possibilities  lie  which  are  seized  upon  and 
woven  into  systems  by  the  different  geometries. 
Under  the  growing  refinement  of  methods  and 
means  of  observation  this  opening  is  a  decreasing 
variable;  it  is  therefore  not  only  conceivable  but 
empirically  possible  that  an  appreciable  deviatioi-Lof 
the  parallel  postulate  from  the  space  of  experience 
may  eventually  be  found.  In  other  words,  as  tested 
by  the  court  of  perceptual  experience  alone  the  case 
may  some  day  go  against  Euclid  but  it  can  never  be 
decided  in  this  way  absolutely  in  his  favor. 

We  reach  then  at  the  close  of  this  chapter  the 
important  conclusion  that  Euclid's  validity  must  be 
established,    if   established    absolutely,    upon    some 
other  basis  'than  that  of  mere  perceptual  experience.  | 
Whether  such  a  basis  exists  remains  to  be  seen. 


THE   NATURE  AND  VALIDITY 


OF  THE 


PARALLEL   POSTULATE. 


CHAPTER  V. 

THE   NATURE  AND  VALIDITY    OF  THE    PARALLEL 

POSTULATE.  / 

We  have  reached  the  conclusion  that  geometrical 
spaces,  merely  as  such,  are  all  of  them  abstract  c(jn- 
ceptions.  They  are  grounded  on  and  grow  out  of 
the  same  general  experience  which  they  interpret 
differently  while  seeking  to  simplify  and  to  system- 
atize it  by  means  of  the  peculiar  postulates  which 
define  them.  We  can  also  see  that  the  quantitative 
concepts  which  underlie  the  different  geometries 
have  been  chosen  somewhat  arbitrarily  so  that  when 
we  carry  them  back  to  the  facts  of  spatial  experi- 
ence they  do  not  reproduce  these  facts  in  any  case 
with  absolute  precision.  Different  groups  of  ideas 
may  therefore  serve  to  express  all  the  facts  with 
equal  exactitude  within  the  region  accessible  to 
observation. 

The  space-world  as  we  know  it  is  not  riuantita- 
tively  infinite  nor  does  the  assumption  that  it  is 
seem  to  be  a  necessary  one.  although  Euclid  requires 
it.  If  then  we  refuse  to  call  it  infinite  in  this  sense 
but  still  accept  the  present  laws  of  optics  and  astron 
127 


128  THE  VALIDITY  OF  EUCLID  • 

omy  which  presuppose  EucHd's  vahdity  and  if  we 
also  admit  the  postulate  of  free  mobility  which  is 
the  same  as  to  assume  that  the  parameter  of  space 
(K)^  is  a  constant  quantity,  we  find  that  even 
under  these  restrictions  it  is  still  possible  to  repre- 
sent all  the  facts  with  equal  accuracy  by  the 
geometries  of  Euclid,  Lobatchewsky,  and  Riemann. 
So  far  then  as  experience  goes  at  present,  or  can 
ever  go  for  that  matter,  there  is  no  necessary  reason 
for  starting  any  physical  inquiry  with  the  Euclidean 
assumption  that  K  is  infinite.  All  we  need  is  to 
take  K  sufficiently  large  to  make  the  deviation  from 
Euclid  fall  within  the  limits  of  astronomical  obser- 
vation. As  this  observation  becomes  more  refined 
and  exact,  one  of  two  things  must  inevitably  hap- 
pen. Either  the  facts  will  ultimately  appear  to  go 
against  Euclid  or  else  it  will  be  shown  that  the 
actually  known  quantity  of  space,  though  always 
finite,  will  transcend  successively  certain  increas- 
ingly large  amounts  which  the  new  approximations 
to  the  value  of  K  will  give  us  the  right  to  affirm. 

But  to  go  so  far  as  to  assert  even  that  K  is  con- 
stant seems,  on  the  surface  at  least,  to  be  an  arbi- 
trary matter  which  is  not  demanded  either  by 
experience  or  by  logic.  It  means  the  same  thing  as 
to  assert  an  absolutely  rigid  standard  which  may 
be   transported   unchanged    to    any   part   of  space. 

1  See  Chapter  II.  under  the  discussion  of  Riemann's  idea  of 
curvature. 


I  HE  VALlUlTy  Of  EUCLID 


ijy 


But  we  certainly  do  not  meet  with  any  such  rigid 
measures  among  the  objects  of  experience  from 
which  as  we  have  seen  the  idea  of  a  metrical  stand- 
ard has  actually  been  derived.  What  we  really 
know  is  that  some  of  these  objects  are  less  subject 
to  quantitative  variations  than  others  are.  and  that 
consequently  a  series  of  them  may  be  arranged 
whose  members  as  we  ascend  the  scale  approximate 
more  and  more  nearly  a  quantitative  invariability. 
Hence  here  as  in  the  case  of  the  straight  line  the 
intellect  may  form  the  pure  abstract  conception  of 
a  rigid  body  which,  when  thought  of  as  independ- 
ent of  position,  furnishes  also  the  conception  of 
spatial  homogeneity.  But  between  the  actual  facts 
of  experience  and  the  essential  proi)erties  of  such 
an  intellectual  construct  there  exists,  as  stated,  a 
gulf  which  sense-perception  of  itself  cannot  bridge. 
It  is  then  both  logically  and  empirically  permis- 
sible to  assume  that  the  actual  parameter  of  space 
is  a  variable  quantity  which  oscillates  within  cer- 
tain narrow  limits.^  By  assuming  this  oscilla- 
tion to  occur  in  accordance  with  law  other  geome- 
tries compatible  with  experience  could  also  be 
obtained. 

It  is  such  considerations  as  these  that  force  upon 

'^  When  we  assume  the  possibility  of  measurement  in  the 
exact  sense  required  by  geometry,  and  the  facts  of  experience 
make  this  of  course  a  legitimate  inference,  this  assumed  varia- 
bility of  K  must  be  ruled  out  on  purely  logical  grounds,  as  we 
shall  later  attempt  to  show. 


130  THE  VALIDITY  OF  EUCLID 

US  the  necessity  of  making  and  maintaining  a  care- 
ful distinction  between  the  facts  of  experience  and 
those  intellectual  constructs  whose  formation  these 
facts  have  suggested.  Upon  these  constructs  two 
conditions  are  imposed :  they  must  fit  the  facts  to 
which  they  relate,  and  must  also  meet  the  logical 
requirements  of  mutual  non-contradiction.  When 
these  conditions  are  fulfilled,  that  is,  when  it  is 
shown  that  different  systems  of  geometry,  Euclid- 
ean and  non-Euclidean,  are  equally  permissible 
under  both  requirements,  the  question  of  validity 
assumes  its  most  interesting  and  difficult  form. 

We  have  shown  that  these  geometries  within  cer- 
tain limitations  already  pointed  out,  are  empirically 
indistinguishable ;  it  therefore  remains  to  consider 
the  other  half  of  the  problem.  Are  these  geometries 
when  considered  strictly  from  the  logical  point  of 
view  equally  tenable?  Fortunately,  as  our  his- 
torical chapter  has  abundantly  shown,  we  already 
have  all  that  seems  to  be  desired  for  a  satisfactory 
answer.  Granting  the  groups  of  assumptions  from 
which  they  set  out,  accepting  the  condition  merely 
that  these  assumptions  be  so  defined  as  to  be  mu- 
tually independent  and  logically  consistent,  and 
finally,  disregarding  the  question  as  to  their  easy 
compatibility  with  the  known  facts  of  reality ; 
almost  an  unlimited  number  of  geometries  can  be. 
and  very  many  indeed  have  been,  actually  built  up 


THE  VALIDITY  OF  EUCLID  131 

in  such  a  manner  as  to  satisfy  the  strictest  demands 
for  internal  consistency. 

When  we  contine  ourselves  then  to  those  consid- 
erations which  the  idea  of  exact  measurement  re- 
quires, and  hence  to  those  assumptions  only  which 
are  in  all  respects  very  similar  to  Euclid's,  the 
demonstrated  fact  that  the  proofs  of  the  resulting 
non-Euclidean  systems  hold  good  for  corresponding- 
theorems  of  Euclid's  geometry  when  the  appro- 
priate substitutions  have  been  made,  shows  beyond 
question  that  any  logical  defects  which  may  possibly 
hereafter  be  discovered  in  any  non-Euclidean  sys- 
tems must  also  apply  with  equal  force  against  Euclid. 
A  revelation  of  contradiction  in  one  must  prove  to 
be  a  revelation  of  contradiction  in  the  other. 

Therefore  the  apparently  inevitable  conclusion 
which  we  are  forced  to  face  is  that  Euclid  is  neither 
empirically  nor  even  logically  necessary  to  the  world 
of  reality.  It  is  based  upon  a  space  conception 
derived  by  abstraction  from  that  world  as  we  know 
it  and  all  its  theorems  and  constructions  are  conse- 
quently in  perfect  logical  harmony  with  that  con- 
ception. But  there  are  other  conceptions  derived 
in  essentially  the  same  way  from  the  same  world 
and  defined  by  slightly  different  postulates.  From 
these  conceptions  also  other  geometries  are  known 
to  flow  with  equal  harmony  and  necessity.  What 
one  then  of  these  opposing  systems  is  the  true  geom- 
etry?    It  seems  almost  absurd  to  ask  this  question. 


132  THE  VALIDITY  OF  EUCLID 

They  are  all,  of  course,  true  —  true  logically  and, 
within  Hmits,  true  to  the  facts;  therefore  the  deci- 
sion between  them,  if  made  at  all,  must  be  made 
not  upon  a  basis  of  truth  but  simply  as  a  matter  of 
convenience.  It  is  not  a  question  of  necessity  but 
one  of  utility.  When  therefore  it  is  generally  ad- 
mitted that  Euclid  is  the  most  convenient  of  them 
all  the  obvious  conclusion  is  that  Euclid  will  con- 
tinue in  favor  though  robbed  of  that  peculiar  maj- 
esty as  a  system  of  absolute  truth  which  it  formerly 
seemed  to  possess. 

But  we  cannot  dispose  of  the  difficulty  in  this 
easy  fashion.  Again  we  have  solved  the  problem  by 
simply  hiding  its  meaning.  The  word  convenient 
here  introduced  is  employed  by  M.  Poincare  as 
though  it  ended  the  matter.  The  axioms  of  geom- 
etry are  for  him  all  of  them  mere  conventions,  they 
are  all  true,  but  some  groups  are  simply  more  con- 
venient  than  others ;  but  in  reality  the  central  philo- 
sophical puzzle  originating  in  metageometry  lies 
concealed  in  Poincare's  use  of  this  word.  We 
cannot  dismiss  the  matter  on  the  mere  ground  of 
convenience.  Granted  that  Euclid  is  the  most  con- 
venient, how,  why,  and  in  what  sense  is  it  so  ?  We 
certainly  cannot  say  that  it  is. because  of  any  ob- 
served matters  of  fact  to  which  geometry  relates, 
nor  of  any  requirement  of  logical  consisteney.  Nor 
can  we  say  that  Euclid  is  logically  the  most  simple. 


I 


THE  VALlUny  Of  EUCLID  ij3 

for  from  this  point  of  view  the  single  elliptic  system 
is  much  more  beautiful  and  attractive. 

It  is  true  that  from  the  standpoint  of  algebra  the 
constructions  of  Euclid  do  not  involve  the  use  of 
such  complicated  equations  as  are  required  by  his 
rivals.  For  these  constructions  nothing  more  diffi- 
cult is  involved  than  a  general  equation  of  the 
quadratic  form.  Hence  the  Euclidean  lines  and 
planes  do  not  require  so  high  a  degree  of  continuity 
as  that  which  is  demanded  by  Lobatchewsky  and 
Riemann.  The  Euclidean  plane  may  be  regarded 
in  fact  as  a  continuous  two-difnensional  manifold 
of  points,  some  of  which  have  been  dropped  out  at 
regular  intervals.  All  the  constructions  of  Euclid 
can  be  made  in  such  a  manifold  without  anywhere 
falling  into  a  hole.  To  sense,  it  would  appear  as  a 
sieve,  and  the  straight 'line  as  a  picket  fence  where 
the  pickets  and  the  distances  between  them  corre- 
spond to  the  points  on  the  line  and  their  relations 
to  each  other.''  This  sort  of  simplicity,  however. 
results  from  the  application  of  numbers  to  luiclidean 
conceptions  and  can  hardly  be  regarded  a?  neces- 
sarily inherent  in  these  conceptions  themselves."'    To 

3  Compare  Dedekind :  IVas  sitid  uud  zcas  sollcn  die  Zahleu. 
Brunswick.  1893-     Vonvort  s.  XII. 

*  Professor  G.  B.  Halsted  states  in  a  letter  to  the  present 
writer  under  date  of  Jan.  4.  1904 :  "  I  build  up  my  Rational 
Geometry  wholly  without  number  or  arithmetic.  usinR  no 
ratios,  irrationals,  or  complexes,  as  you  will  see  when  it  ap- 
pears in  a  month  or  two:" 


134  THE  VALIDITY  OF  EUCLID 

say  that  Euclid  is  the  most  convenient  in  this  sense 
is  but  to  say  that  his  conceptions  happen  to  be  most 
amenable  to  easy  algebraic  interpretation.  But  this 
is  certainly  no  mere  accident.  Like  every  other  fact 
of  human  experience  it  demands  philosophic  recog- 
nition. Why  indeed  should  it  be  that  the  very  con- 
ception of  space  which  is  historically  first  and  at  the 
same  time  most  natural,  should  also  be  the  most 
easily  interpretable  by  means  of  algebraic  formulae 
which  have  been  derived  from  a  widely  different 
and  apparently  independent  source  in  experience? 
Why  does  that  which  is  spatially  the  most  simple  and 
which  can  be  successfully  handled  without  regard 
to  number,  thus  prove  to  be  at  the  same  time  most 
convenient  for  numerical  interpretation? 
i  In  attempting  to  answer  this  question  the  influence 
)of  custom  should  not  be  ignored.  Relations  which 
have  been  established  by  long  and  uninterrupted  asso- 
ciations are  not  easily  distinguished  from  those  which 
are  logically  necessary.  Hence  it  must  not  be  for- 
gotten that  civilization  has  been  steeped  in  Euclid 
for  more  than  two  thousand  years  so  that  today  this 
geometry  underlies  all  our  physical  sciences  and  we 
are  giving  expression  to  Euclidean  forms  in  all  the 
mechanical  arts. 

During  this  long  period  the  reasoning  and  even 
the  form  of  Euclid  have  been  generally  regarded  as 
the  most  perfect  model  of  scientific  and  even  of  philo- 


IHE  rAUDITY  OF  EUCLID  135 

sopliic '"'  thought  and  expression.  (Jur  intellectual 
life  has  become  attuned  both  to  Euclid's  doctrine  of 
space  and  to  his  peculiar  form  of  expression.  Little 
wonder  then  that  it  seems  almost  sacrilege  to  depart 
from  them  or  even  to  call  them  in  question ;  we  come 
to  feel  with  Kant  and  with  thousands  of  others,  tiiat 
if  there  is  an  a  priori  necessity  anywhere  in  human 
knowledge  we  find  it  in  Euclid.  But  suppose  all  this 
had  not  been,  and  there  is  surely  nothing  violent  in 
this  supposition,  could  we  still  say  that  Euclid  is  the 
most  convenient  geometry  ? 

We  have  seen  how  the  peculiar  facts  which  have 
pointed  unmistakably  to  Euclid  have  been  the  most 
patent,  universal,  and  familiar  at  all  stages  of  the 
race's  development.  They  have  not  needed  pro- 
longed meditation,  observation,  and  experiment  to 
bring  them  into  prominence,  they  lie  open  and  every- 
where ready  at  hand.  In  consequence,  Euclid  has 
gained  a  long  start  in  advance  of  his  competitors  in 
the  race  for  general  acceptance,  and  popular  favor. 

But  here  again  we  must  face  the  question  which 
seems  to  confront  us  withersoever  we  turn.     Why 


''  Spinoza's  Ethics  is  a  good  illustration  of  Euclid's  inflncnce 
in  this  direction.  Even  the  fundamental  ontolopical  concep- 
tions have  often  been  borrowed  directly  or  indirectly  from 
Euclid.  This  veneration  of  Euclid  as  a  body  of  ideal  knowl- 
edge has  proved  exceedingly  mischievous  in  many  ways. 
YMathematics  and  formal  logic  instead  of  being  the  very  ideal 
of  truth  are  in  an  important  sense  the  farthest  removed  from 
truth. 


136  THE  VALIDITY  OF  EUCLID 

this  ancient  and  more  favorable  start  ?  It  is  certainly 
not  a  mere  matter  of  chance  that  our  world  of  experi- 
ence should  be  so  favorably  disposed  for  the  sugges- 
tion of  Euclid.  Though  the  mere  antiquity  of  this 
science,  in  itself,  is  no  proof  of  its  absolute  and  ex- 
clusive necessity;  no  sure  demonstration  that  it,  as 
against  other  logically  and  empirically  justifiable 
possibilities,  is  alone  a  priori  and  native  to  the  con- 
stitution of  the  mind;  nevertheless  it  does,  when 
taken  in  conjunction  with  the  matter  of  algebraic 
simplicity  already  pointed  out,  compel  us  to  consider 
Kant's  doctrine.  There  is  certainly  something, 
either  in  the  knowing  mind  or  the  world  that  is 
known,  which  makes  for  Euclid,  and  philosophy  is 
called  upon  to  locate  it  and  to  determine  satisfac- 
torily its  essential  nature. 

Is  Euclid  then,  as  Kant  maintained,  based  on  an 
a  priori  form  of  our  sense  intuition  ?  Must  we  see 
things  whether  we  will  or  no,  through  Euclidean 
glasses?  And  are  the  non-Euclidean  systems  after 
all  merely  ingenious  and  interesting  intellectual  con- 
structs which  cannot  even  be  thought  of  as  realized 
except  in  Euclidean  spaces? 

The  first  of  these  questions,  in  the  light  of  our 
previous  discussion,  is  now  comparatively  easy  to 
answer;  the  last  will  be  reserved  for  the  following 
chapter.  At  present  then  we  shall  have  nothing  to  * 
do  with  those  spatial  characteristics  which  Euclid 
and  non-Euclid  hold  in  common  with  each  other. 


THE  VALIDITY  OF  EUCLID 


ii7 


We  are  concerned  merely  with  that  pecuHar  feature 
which  distinguishes  EucHd.  Dehn's  investigation 
has  put  beyond  question  what  Euchd's  one  essential 
pecuHarity  is;  our  present  task  therefore  concerns 
merely  the  parallel  postulate.  Can  it  be  maintained 
that  this  postulate  is  a  priori  in  Kant's  meaning  of 
that  term?  If  it  cannot,  then  Euclid's  last  claim  of 
necessary  supremacy  must  be  rejected. 

By  space  Kant  meant,  of  course,  a  universal  and 
necessary  form  of  sense-intuition  which  is  native  to 
the  mind.  By  establishing  this  thesis  as  a  firm 
ground  of  standing  he  sought  to  explain  how  it  is 
possible  to  have  a  knowledge  of  objects  not  only 
prior  to  all  experience  with  them  but  which  tran- 
scends the  very  bounds  of  all  possible  experience. 

Kant  never  raises  the  question  as  to  whether  a 
priori  synthetic  knowledge  of  any  sort  is  possible. 
"  Of  course  it  is  possible,"  he  would  say.  "  in  fact, 
it  is  actual,  for  we  have  it  already  in  Euclid."  In- 
deed Euclid  in  Kant's  day  was,  and  had  been  for 
centuries,  universally  accepted.  Hence  to  prove  be- 
yond cavil  that  certain  knowledge  of  the  real  world 
was  possible  independent  of  experience  Kant  had 
but  to  point  to  Euclid.  There  geometry  as  a  beauti- 
ful science  stopd,  that  it  had  apodeictic  certainty 
could  not  be  doubted,  for  "  none  but  a  fool  could 
doubt  its  validity  or  deny  its  objective  reference." 
An  a  priori  synthetic  body  of  knowledge  is  therefore 
possible:  but  how  is  it  possible?     That  was  Kant's 


138  THE  VALIDITY  OF  EUCLID 

problem.  If,  as  he  thought,  Euchd  has  apodeictic 
certainty  then  space  must  be  a  priori  and  purely  sub- 
jective; conversely,  if  space  is  subjective  geometry 
must  have  apodeictic  certainty.  Hence  his  argument 
assumes  a  twofold  form.  On  the  one  hand  geometry 
exists  as  a  science  and  is  known  to  have  apodeictic 
certainty;  therefore  it  follows  that  space  is  a  priori 
and  subjective.  On  the  other  hand,  it  follows  from 
considerations  which  are  independent  of  geometry 
that  space  is  subjective  and  a  priori,  therefore  geom- 
etry must  have  apodeictic  certainty. 

We  now  purpose  to  show  that,  in  the  light  of  our 
previous  discussion,  the  first  of  these  two  arguments, 
taken  by  itself,  falls  short  of  the  mark,  and  makes 
against  Euclid ;  and  that  the  second  is  just  as  valid 
for  non-Euclidean  space-forms  as  it  is  for  Euclid's 
and  applies  in  so  far  as  it  can  be  accepted  at  all  only 
to  what  these  space-forms  possess  in  common.  The 
position  which  we  shall  thus  aim  to  establish  is  that 
the  parallel  postulate  is  not  a  priori  but  empirical  in 
character.  , 

Kant's  first  argument,  in  so  far  as  it  is  distinct 
from  the  second,  infers  from  the  mere  existence  of 
Euclid  the  a  priori  and  subjective  character  of  Eu- 
clidean space.  Prior  to  the  introduction  of  non- 
Euclidean  geometry  this  argument  seemed  of  course 
to  be  a  forcible  one,  but  now  the  case  is  quite  dififer- 
ent.  For  if  we  do  not  beg  the  whole  question  at  the 
outset  by   dogmatically   asserting  that   Euclid   has 


THE  VALIDITY  OF  EUCLID 


U9 


intuitive  certainty  while  non-Euclid  has  not,  and  if 
we  also  accept  the  position  already  established  that 
these  geometries  are  otherwise  on  a  par  with  each 
other,  it  must  be  admitted  that  Kant's  argument 
applies  as  well  to  the  modern  systems  as  it  does  to 
Euclid.  But  when  the  argument  is  thus  extended 
its  invalidity  becomes  at  once  apparent.  For  if  we 
assume  three  a  priori  space  forms  corresponding  to 
these  three  geometries,  and  we  dare  not  now  make 
this  assumption  with  regard  to  one  without  also 
making  it  with  regard  to  the  others,  they  enter  into 
a  hopeless  conflict  with  each  other.  For  if  it  must 
be  universally  and  necessarily  true  that  the  sum  of 
the  angles  of  a  triangle  is  exactly  equal  to  two  right 
angles  it  cannot  also  be,  at  the  same  time,  uni- 
versally and  necessarily  true  that  this  same  sum  is 
also  greater  and  less  than  two  right  angles  accord- 
ing as  we  happen  to  be  speaking  of  Euclidean,  Lo- 
batchewskian,  or  Riemannian  space. 

The  mere  existence  of  Euclidean  geometry  is 
therefore  not  in  itself  sufificient  to  prove  the  possi- 
bility of  an  a  priori  intuition  of  Euclidean  space. 
Logically  this  is  a  distinction  which  can  no  more  be 
claimed  for  Euclidean  than  for  non-Euclidean 
geometry.  We  now  see  that  it-  cannot  be  claimed 
for  both  without  contradiction,  therefore  it  cannot 
be  claimed  for  either,  and  this  division  of  Kant's 
argument  falls  to  the  ground. 

But  in  proving  the  invalidity  of  this  one  argu- 


140  THE  VALIDITY  OF  EUCLID 

ment  we  have  not  done  with  Kant.  In  the  fore- 
going refutation  we  have  made  use  of  the  principle 
of  contradiction  to  establish  our  position ;  but  Kant 
denies  that  this  principle  applies.  He  claims  that 
we  can  frame  an  intuition  of  Euclidean  space  and 
that  a  priori.  On  page  20,  for  instance,  of  Muller's 
Translation  of  the  Critique  of  Pure  Reason,  Kant 
affirms  that  all  geometrical  principles  as,  for  ex- 
ample, "  that  in  every  triangle  two  sides  are  to- 
gether greater  than  the  third  side,"  are  never  to  he 
derived  from  general  concepts  of  side  and  triangle 
but  from  an  intuition,  and  that  a  priori,  with  apo- 
deictic  certainty.  Kant  here  sets  up  the  claim  that 
geometrical  reasoning  is  not  fundamentally  a  mere 
matter  of  logical  consistency,  but  by  virtue  of  our 
supposed  intuition  of  space,  it  is  synthetic  and  can- 
not, though  a  priori,  be  upheld  by  the  principle  of 
contradiction  alone. 

Admitting  the  soundness  of  this  claim  for  the 
present,  let  us  examine  the  arguments  which  Kant 
advances  to  support  it.  These  are  embodied  in  his 
general  doctrine  of  the  a  priori  synthetic  nature  of 
geometrical  judgments  and  also  in  the  five  argu- 
ments given  on  pages  18-20  of  Muller's  translation. 
The  latter  may  be  omitted  here  with  the  simple 
remark  that  if  admitted  at  their  face  value  they 
prove  nothing  peculiar  to  the  special  nature  of 
Euclidean  space.  When  taken  into  full  confidence 
they  only  establish  the  necessity  of  space  as  a  form 


THE  VALIDITY  OF  EUCLID  141 

of  externality,  a  sort  of  differentiating  principle 
whose  special  nature  is  left  undetermined.  For  our 
purpose  then  we  need  only  consider  the  argument 
from  the  a  priori  synthetic  nature  of  geometrical 
judgments.  Kant  held  that  these  judgments  are 
not  deducible  from  logic;  their  contradictories  are 
not  self-contradictory.  They  combine  subjects  with 
predicates  which  cannot  be  shown  by  logic  to  have 
any  connection  whatever,  and  yet  these  judgments 
have  apodeictic  certainty.  They  are  apodeictic  not 
simply  because  we  have  a  subjective  conviction  that 
they  are,  but  because  witliout  them  experience  itself 
would  be  impossible.    Let  us  examine  this  claim. 

Since  Kant's  day  attention  has  been  called  to  the 
fact  that  any  judgment  may  be  either  analytic  or 
synthetic  according  to  the  point  of  view  from  which 
we  consider  it.^  While,  therefore,  this  distinction 
so  sharply  drawn  and  so  greatly  emphasized  by 
Kant  has  a  logical  value  as  marking  predominant 
aspects  rather  than  exclusive  characters  in  the  classi- 
fication of  judgments  it  is  really  without  value  for 
a  theory  of  knowledge.  For  if  every  judgment  is 
analytic  in  one  aspect,  at  least,  the  principle  of  con- 
tradiction may  be  applied  to  all  judgments  without 
exception,  and  our  previous  argument  hold-. 

To  make  our  meaning  clear  let  us  take  Kant's 
own  example  of  a  synthetic  judgment :  "  All  bodies 

«  Consult  Bosanquet's  Logic.  Vol.  I.,  pp.  Q7  to  103 


142  THE  VALIDITY  OF  EUCLID 

are  heavy."  This  judgment  he  claims  is  purely 
synthetic,  but  is  it  not  also  analytic?  If  the  subject 
signifies  one's  developed,  every-day  "motion  of  body, 
the  judgment  is  obviously  analytic,  for  weight  is  as 
much  an  inseparable  attribute  of  this  notion  of  body 
as  is  extension  itself.  If  then  this  judgment  is  to 
be  regarded  as  synthetic  we  must  go  back  to  the 
origin  of  the  concept  body  and  ask  how  it  was  that 
the  attribute  weight  came  to  be  joined  to  it.  If 
one  had  no  muscular  sense,  which  alone  acquaints 
one  with  facts  of  resistance,  his  conception  of  body 
would  never  involve  the  attribute  weight;  but  if  he 
came  somehow  to  be  suddenly  possessed  of  this 
sense  and  through  actual  experience  with  bodies 
became  conscious  of  weight,  the  judgment,  "  All 
bodies  are  heavy,"  would  for  him  be  synthetic,  for 
it  adds  a  new  predicate  to  his  concept  body. 

But  on  the  other  hand  let  us  suppose  a  being  born 
into  the  world  in  possession  of  all  the  special  senses 
except  sight  and  touch.  For  such  an  hypothetical 
being  the  judgment  "  all  bodies  are  heavy  "  would 
certainly  be  analytic  for  weight  must  be  for  him  an 
inseparable  attribute  of  body.  If  now  we  restore 
the  lost'  senses  of  sight  and  touch  and  allow  him  to 
see  and  handle  objects  until  through  experience  he 
is  able  to  form  the  judgment  which  Kant  calls  an- 
alytic, "  All  bodies  are  extended,"  it  is  plain  that 
for  him  such  a  judgment  would  appear  to  be  syn- 


THE  VALlDnV  OF  EUCLID  143 

thetic,  for  as  before  it  adds  in  the  predicate  some- 
thing not  previously  found  to  be  in  the  subject. 

But  it  may  be  objected  that  Kant's  distinction 
between  analytic  and  synthetic  judgments  was  not 
based  upon  the  origin,  but  upon  the  nature,  of  the 
connection  between  subject  and  predicate.  In  reply 
we  need  to  discriminate  between  the  essential  nature 
of  the  judgment  and  its  symbolic  expression.  Fail- 
ure to  distinguish  the  judgment  itself  from  one's 
mode  of  apprehending  it  often  occasions  the  error 
of  regarding  as  purely  synthetic  a  judgment  which 
is  in  fact  analytic.  If  we  but  attend  to  the  nature 
of  the  judgment  itself  regardless  of  its  external 
expression  and  how  we  came  to  know  it.  the  rela- 
tion between  subject  and  predicate  is  seen  to  be  an 
identical  one.  This  we  believe  to  be  true  of  geo- 
metrical judgments  in  general;^  the  principle  of 
identity  rules  throughout.  This  becomes  evident 
when  we  consider  the  way  in  which  such  judgments 
are  actually  extended.  This  extension  does  not  in- 
volve d  necessary  synthesis ;  nor  does  it  depend  upon 
the  universality  and  necessity  of  space  as  an  a  priori 
form  of  sense-intuition,  but  rather  upon  the  iden- 
tity and  homogeneity  of  space  as  an  abstract  con- 
ception. It  is  therefore  analytic  and  follows  as  a 
natural  consequence  from  this  homogeneous  nature 
of  space.     All  the  judgments  of  physics  are  hypo- 

7  Consult  Lacld's  A  Thcnr\   of  Reality,  pp.  3.^1   ff 


144  THE  VALIDITY  OF  EUCLID 

thetically  a  priori,  that  is  to  say,  the  connection  be- 
tween subject  and  predicate  is  strictly  necessary, 
provided  we  may  be  allowed  to  assume  that  the  sub- 
ject remains  always  identically  the  same.  But  in 
this  science  the  subject  as  a  rule  does  not  meet  this 
requirement,  for  as  a  result  of  further  investigations 
it  is  liable  to  change.  In  space,  however,  we  have 
something  which  is  homogeneous  and  identical 
always  and  throughout;  so  that  in  it,  because  of 
this  peculiar  character,  such  a  thing  as  difference 
or  change  cannot  occur.  Consequently  all  geometri- 
cal judgments,  which  do  but  exploit  the  nature 
of  space,  flow  directly  from  this  concept  by  the 
principle  of  identity.  But  this  identity  or  homo- 
geneity of  space  as  we  have  already  shown  is  not  an 
intuition  but  a  conceptual  property.  Geometrical 
judgments  are  dependent  on  the  nature  of  the  space 
concept,  and  simply  define  its  content.  These  judg- 
ments are  therefore  analytic  in  nature  and  conse- 
quently, here  as  elsewhere,  the  principle  of  contra- 
diction may  be  applied. 

Euclid's  judgments,  therefore,  are  not  something 
sui  generis,  nor  is  this  particular  space-form  proved 
by  Kant  to  be  an  intuitive  necessity  presupposed 
in  the  very  possibility  of  sense-experience;  nor  is 
Euclidean  reasoning  possessed  of  any  peculiar  cer- 
tainty that  can  be  denied  to  non-Euclidean  systems. 
Geometrical  certainty  of  any  sort,  Euclidean  or  non- 
Euclidean,  is  but  the  certainty  with  which  any  con- 


THE  VALIDITY  OF  EUCLID  145 

elusions  follow  from  non-contradictory  premises. 
It  is  logical  certainty  only,  and  in  each  case  flows 
directly  from  definition.  The  certainty  with  which 
the  sum  of  the  angles  of  any  triangle  may  be  as- 
serted to  equal  two  right  angles  in  Euclidean  geom- 
etry is  exactly  the  same  sort  of  certainty  as  that  by 
which  it  may  be  shown  to  be  more  or  less  than  two 
right  angles  in  the  other  two  geometries.  In  each 
case  it  is  but  the  certainty  of  intrinsic  consistency. 
Kant  has  certainly  failed  to  prove  the  a  priori 
necessity  of  Euclidean  space.  Therefore  the  objec- 
tion often  raised  that  we  have  no  intuition  of  non- 
Euclidean  space-forms  comes  without  force  against 
these  systems  until  some  one  has  shown  that  Euclid 
is  in  any  special  sense  intuitively  certain. 
/  Our  conclusion  therefore  is  that  the  parallel  pos-j 
Ltulate  is  not  a  subjective  necessity ;  it  is  not  essential 
to  the  constitution  of  man's  intellectual  nature  in 
the  strict  sense  in  w^hich  Kant  claimed  it  to  be  so. 
There  remains  nothing,  so  far  as  we  can  see.  on 
which  such  a  claim  may  be  grounded  except  the 
mere  subjective  conviction  that  it  must  be  so;  ob- 
viously this  conviction  cannot  count  for  much  in 
the  case  of  one  who  simply  does  not  feel  it.  Fur- 
thermore any  such  extreme  doctrine  of  the  exclu- 
sive subjectivity  of  space  defeats  its  own  ends.* 
For  if  space  is  thus  merely  a  private  form  of  intu 

8  Consult  Professor  Ladd's  A  Theory  of  Reality:  the  chap- 
ter on  "Space  and  Motion,"  especially  pages  235-240. 


146  THE  VALIDITY  OF  EUCLID 

ition,  having  no  correlate  in  objective  reality,  it  is 
certainly  possible  that  other  beings  may  have  other 
space-forms  without  suspecting  any  difference  in 
the  world  which  they  arrange  under  them.  There- 
fore Euclid's  validity,  instead  of  being  shown  in 
this  way  to  be  a  necessity,  could  only  be  estab- 
lished by  an  empirical  investigation  of  the  nature 
of  the  actual  forms  of  space-intuition  of  a  large 
number  of  individuals.*^ 

Let  us  go  back  now  to  answer  the  question  which 
led  to  this  long  consideration.  The  position  is  now 
certainly  justified  that  the  comparative  simplicity 
and  convenience  of  Euclid  is  not  due  to  anything 
essential  to  intelligence  as  such,  nor  is  the  fact  that 
Euclid  is  more  amenable  to  algebraic  interpretation 
than  his  modern  rivals  to  be  regarded  as  a  necessity 
of  any  rational  nature.  That  certain  algebraic  for- 
mulae possess  greater  simplicity  than  others  and  that 
^  Euclid  as  a  system  builds  itself  up  in  the  mind  more 
easily  and  readily  from  certain  points  of  starting 
than  from  others  are  facts  which  seem  native  to  our 
minds  in  a  much  truer  sense  than  the  ordinary  facts 
of  sense-perception ;  but  that  these  facts  are  a  priori 
in  the  sense  of  being  necessary  presuppositions  of 
rationality  itself  and  therefore  essential  to  the  very 
possibility  of  any  world  whatever  must  certainly  be 
denied.    For  from  this  purely  rational  point  of  view 

9  Compare  Lotze's  Metaphysics,  Book  II.,  Chap.  2. 


THE  lALlDlTV  01-  EUCLID 


H7 


Strictly  taken  there  exists  no  reason  why  one  point 
of  setting  out  should  be  regarded  as  better  or  more 
convenient  than  another.  The  rational  goal  in  each 
case  is  simply  to  construct  the  entire  system,  and 
when  a  sufficient  number  of  conditions  are  postu- 
lated to  define  completely  what  is  logically  involved, 
no  room  for  a  preference  is  left.  No  difference 
being  admitted  in  the  nature  of  the  entities  em- 
ployed none  could  be  allowed  in  the  relations  be- 
tween them,  and  it  ought  to  be  just  as  simple  and 
convenient  to  trace  these  relations  from  one  point 
of  starting  as  from  another. 

It  is  not  meant,  of  course,  that  no  more  than  this 
is  essential  to  a  world  as  actually  known,  nor  is  it 
maintained  that  a  merely  formal  world  could  really 
exist  or  be  truly  known  if  it  did  exist.  That  a 
world  of  real  beings  each  with  peculiar  forms  and 
laws  of  its  own  must  be  postulated  in  the  interests 
of  a  genuine  cognition  will  be  readily  granted ;  that 
they  are  is  a  presupposition  necessary  to  the  possi- 
bility of  experience,  but  precisely  z^hat  they  are  in 
particular,  is  a  matter  which  experience  itself  must 
reveal.  Hence  just  what  algebraic  formulae,  what 
geometry,  and  what  points  of  starting  in  any  par- 
ticular geometry,  are  really  most  convenient  and 
simple  cannot  be  foretold  by  any  mere  analysis  of 
what  it  means  to  be  rational.  These  are  truths 
which  are  essentially  a  postcriorc  in  their  character: 
they  come  from   experience,  individual  and   racial. 


148  THE  VALIDITY  OF  EUCLID 

and  hence  to  experience  itself  the  ultimate  appeal 
must  be  made  to  tell  what  they  are. 

We  may  certainly  conclude  then  that  the  parallel 
postulate  and  the  corresponding  assumptions  of  the 
other  geometries  are  competitive  possibilities  aris- 
ing out  of  the  difference  between  the  exactitude 
demanded  by  mathematics  and  the  approximate  pre- 
cision which  the  limitations  of  sense-perception  in- 
evitably impose.  The  question  as  to  which 
possibility,  if  any  one  of  the  group,  exactly  accords 
with  reality,  is  one  which  can  be  answered  only 
when  the  nature  of  reality  is  itself  more-  accurately 
known.  The  validity  of  Euclid  is  therefore,  at  bot- 
tom, an  empirical  matter  which  can  be  decided,  so 
far  as  decision  is  possible,  only  by  a  more  refined 
appeal  to  actual  sense-experience. 

The  scientific  question  as  to  how  this  appeal  can 
best  be  made  has  frequently  come  to  the  surface  in 
what  has  been  said,  but  a  fuller  reply  to  it  must  now 
be  attempted.  And  first  let  account  be  taken  of 
the  peculiar  difficulties  which  beset  even  the  best 
evidence  we  have  concerning  the  nature  of  actual 
space.  The  Euclidean  conception  is  approximately 
correct.  This  we  know.  The  facts  thus  far  do  not 
necessitate  the  least  departure  from  it,  and  yet  it 
is  possible,  though  not  very  probable,  that  such 
anomalous  facts  may  at  length  be  found.  But  if 
they  should  be  found,  there  is  every  consideration 
to  dispose  one  to  interpret  them,  if  possible,  in  har- 


THE  VALIDITY  Of  EUCLID  149 

mony  with  Euclid.  In  the  actual  nieasurenients 
of  astronomical  distances,  and  in  the  formulation 
of  the  laws  of  optics,  physics  and  astronomy  the 
parallel  postulate  has  been  confidently  accepted.  It 
is  furthermore  a  custom  with  physicists  resulting 
from  long  and  thoroughly  tried  practical  experi- 
ence, to  adhere  steadfastly  to  the  simplest  assump- 
tions until  the  facts  have  so  complicated  them  as 
to  force  their  rejection  or  their  modification.  Now 
of  all  the  concepts  which  any  physical  inquiry  must 
employ,  the  simplest  pf  course  are  those  of  space 
and  time.  We  can  imagine  any  sort  of  figure  or 
complication  of  figures  to  be  realized  in  Euclidean 
space  without  doing  violence  to  this  conception. 
Other  concepts  are  more  narrowly  restricted  by  the 
nature  of  the  facts,  A  perfect  gas,  or  a  perfectly 
elastic  body,  for  example,  does  not  exist.  The 
physicist  is  perfectly  conscious  that  these  are 
fictions  which  conform  only  approximately  and  by 
arbitrary  simplifications  to  the  actual  facts.  There 
are  deviations  which  cannot  be  removed.  Hence. 
among  the  conceptions  of  physics  there  is  a  recog- 
nized scale  of  perfection :  and  when  facts  are  dis- 
covered that  render  a  modification  of  some  sort 
necessary  it  is  naturally  the  less  perfect  conceptions 
which  must  suffer  the  change.  Consefpiently  if 
facts  should  be  discovered  which  seemingly  go 
against  Euclid,  it  would  be  simpler  and  easier  to 
maintain   that  the  laws   of  physics,   astronomy   or 


150  THE  VALIDITY  OF  EUCLID 

optics  are  slightly  incorrect  than  to -sacrifice  Euclid 
and  reorganize  the  whole  of  physical  science  upon 
the  basis  of  a  different  geometry. 

It  is  therefore  obvious  that  mere  measurements 
of  stellar  parallax,  however  refined,  can  never  be 
regarded  as  conclusive  evidence  either  for  or  against 
Euclid.  Under  present  instrumental  limitations 
and  since  the  base  line  cannot  in  any  event  be  greater 
than  the  diameter  of  the  earth's  orbit,  the  probable 
error  of  the  best  astronomical  observations  admits 
the  possibility  of  a  departure  of  a  triangle's  angle 
sum  from  two  right  angles,  amounting  to  ten  de- 
grees or  more  in  the  case  of  stellar  triangles  whose 
sides  all  equal  the  distance  from  the  earth  to  Sirius. 
Such  a  measurement,  assuming  as  it  does  that  rays 
of  light  from  the  most  distant  stars  are  Euclidean 
straight  lines,  is  far  from  proving  that  K  is  infinite. 
It  only  shows  that  the  actual  space  constant  is  very 
large  as  compared  even  with  stellar  distances. 

For  short  measurements  on  the  earth  rays  of 
light  are  perhaps  the  best  examples  we  have  of 
actual  straight  lines,  and  we  naturally  assume  that 
they  maintain  this  characteristic  even  in  the  most 
remote  regions  of  space.  We  must  make  some 
assumption  regarding  the  course  of  light  in  these 
far  off  regions,  otherwise  astronomical  measure- 
ments are  altogether  impossible.  In  doing  so  we 
are  obliged  to  infer  that  what  holds  true  of  light 
here  also  holds  true  of  it  there,  but  we  have  seen 


THE  VALIDITY  OF  EUCLID 


J51 


that  within  the  hmits  of  fact  which  must  serve  as 
a  basis  for  this  inference  Euclidean  and  non-Euclid- 
ean straight  lines  are  not  to  be  distinguished  from 
each  other.  Hence  with  K  finite  but  very  large 
as  compared  with  terrestrial  measurements  and 
with  different  laws  of  optics  the  same  facts  might 
be  accounted  for  in  a  manner  quite  as  simple  as  they 
are  at  present. 
^  On  the  other  hand,  the  discovery  of  a  parallax 
in  the  case  of  the  most  distant  star  or  of  a  negative 
parallax  for  any  star  cannot  be  regarded  as  disprov- 
ing Euclid.  Such  anomalies,  if  not  too  numerous, 
could  be  accounted  for  in  a  manner  which  is  much 
less  expensive  by  making  suitable  changes  in  the 
received  laws  of  optics;  or  even  by  assuming  a 
slight  strain  in  the  ether  itself  in  those  particular 
regions  of  space.  While  therefore  an  actual  dis- 
covery of  any  such  facts  would  certainly  prove 
interesting  and  suggestive  this  alone  could  not  be 
accepted  as  a  conclusive  proof  or  disproof  of 
Euclid's  validity;  for  the  very  conditions  which 
would  render  this  discovery  possible  must  be  found 
to  involve  certain  assumptions  which  might  very 
well  be  withdrawn.  The  facts,  therefore,  which 
count  most  either  for  or  against  Euclid  must  be 
found  if  possible  within  the  realm  of  direct  obser- 
vation where  such  assumptions  as  to  what  takes 
place  beyond  this  realm  do  not  need  to  be  made. 
The  problem  then  is  at  bottom  a  psychological  one 


iSa  THE  VALIDITY  OF  EUCLID 

which  must  be  decided  for  the  most  part  by  experi- 
ment. What  are  the  sensory  contributions  which 
should  be  taken  into  account  in  the  formation  of 
the  abstract  conception  of  space?  If  we  confine  our- 
selves to  the  contributions  of  vision  alone  our  geom- 
etry is  not  Euclidean ;  it  is  projective,  and  so  far  as 
its  space  conception  is  concerned  is  not  to  be  dis- 
tinguished from  the  double  or  single  elliptic  systems 
of  metrical  geometry.  But  if  we  consider  the  larger 
realm  of  spatial  experiences  in  which  sensations  of 
motion,  direction  and  touch  are  also  involved,  all 
the  known  facts  inevitably  suggest  the  parallel  pos- 
tulate. Whether  they  shall  continue  to  do  so  as 
tested  by  the  most  refined  experimental  analysis  of 
man's  ability  to  discriminate  under  the  most  favor- 
able conditions  slight  variations  in  the  size  of  angles, 
in  the  length  of  lines,  and  in  the  latter' s  departure 
from  ideal  straightness  remains  to  be  determined. 

As  already  stated,  the  author  has  undertaken  an 
experimental  investigation  of  this  sort,  at  the  sug- 
gestion of  Professor  Ladd.  Unfortunately  the 
data  thus  far  obtained  are  not  sufficient  to  justify 
any  positive  statement  as  to  what  the  outcome  will 
probably  be.  Obviously  a  very  great  number  and 
variety  of  experiments  need  to  be  performed  by  a 
large  number  of  individuals  to  obtain  results  that 
can  be  regarded  as  significant. 


I 


CONCLUSIONS 


AS    TO    THE 


NATURE   OF  SPACE. 


CHAPTER    VL 

RESULTING    IMPLICATIONS    AS    TO    THE    NATURE    OF 
SPACE. 

In  the  present  chapter  it  shall  be  our  purpose 
to  point  out  certain  imphcations  as  to  the  nature 
of  space  which  seem  to  result  from  denying  the 
necessary  validity  of  the  parallel  postulate  and  from 
the  consequent  possibility  of  non-Euclidean  geome- 
tries. As  a  preparatory  step  in  this  direction  it  is 
necessary  to  get  a  clear  understanding  of  precisely 
what  is  meant  by  non-Euclidean  spaces.  The 
question  therefore  recurs,  are  these  so-called  spaces 
after  all  anything  more  than  ingenious  logical  con- 
structs which  one  cannot  even  think  of  except  as 
tinite  forms  in  Euclidean  spaces? 

In  framing  a  reply  to  this  question  we  must 
guard  against  certain  errors  which  almost  inevit- 
ably result  from  the  introduction  of  Euclidean 
analogies  of  corresponding  non-Euclidean  concep- 
tions. Spherical  and  pseudo-spherical  surfaces 
upon  which  non-Euclidean  straight  lines  arc  repre- 
sented as  curves  in  ordinary  Euclidean  space  of 
three  dimensions  are.  in  reality,  very  diflfcrent  from 
155 


156  THE  NATURE  OF  SPACE 

the  corresponding  non-Euclidean  two-dimensional 
spaces  which  they  are  designed  to  represent.  These 
surfaces  are  not  non-Euchdean  spaces,  although  the 
analytical  treatment  is  the  same  in  both  cases. 
They  are  merely  analogies  to  aid  the  imagination. 
Failure  to  understand  this  distinction  has  often  led 
to  the  error  of  regarding  non-Euclidean  two-dimen- 
sional spaces  as  curved  surfaces  in  tri-dimensional 
Euclidean  space,  and  non-Euclidean  spaces  of  three 
dimensions  as  constructs  requiring  a  Euclidean 
space  of  four  dimensions  for  their  possible  realiza- 
tion. 

If  this  view  were  correct  it  would  be  very  easy 
to  solve  the  special  philosophical  problem  by  reduc- 
ing everything  to  Euclid.  A  non-Euclidean  space 
of  any  number  of  dimensions  gets  itself  realized  in 
a  Euclidean  space  of  higher  dimensions,  non-Eu- 
clidean planes  are  Euclidean  curved  surfaces,  and 
non-Euclidean  straight  lines  are  Euclidean  curves; 
hence  all  we  need  is  a  vocabulary  whereby  the  lan- 
guage of  one  system  may  be  translated  into  that  of 
another,  and  the  problem  is  very  beautifully  solved. 
Non-Euclidean  geometry  thus  becomes  a  mere  play 
upon  words ;  it  is  simply  Euclid  with  a  change  of 
names.  The  next  step  is  to  argue  that  Euclidean 
spaces  of  four  or  more  dimensions  are  inconceivable, 
and  consequently  that  non-Euclidean  spaces  of  even 
three  dimensions  are  impossible.  This  done,  the 
refutation  of  non-Euclid  would  stand  complete. 


^ 


UNIVERSITY 
_       or 


THE  NATURE  OF  SPACE  157 

But  this  solution  of  the  clifficuky  would  be  alto- 
gether too  easily  won.  There  seems  to  be  an  error 
involved,  and  we  need  to  discover  its  source.  The 
confusion  springs,  it  seems  to  me,  from  a  misappre- 
hension of  the  proper  meaning  of  space  and  of  curva- 
ture as  applied  to  space.  To  dispel  the  fog  we  must 
clear  up  the  meaning  of  these  two  conceptions. 

Space,  properly  speaking,  always  means  a  total- 
ity. There  is  considerable  difference  between  a  Eu- 
clidean tw^o-dimensional  space,  for  example,  and  the 
Euclidean  plane  as  ordinarily  conceived.  Their  in- 
ternal relations  are  exactly  the  same ;  their  analytical 
treatment  is  also  the  same.  But  the  plane  has  in 
addition  to  these  internal  relations  certain  external 
relations ;  it  is  two-sided,  for  instance,  and  has  posi- 
tion. A  two-dimensional  space,  on  the  other  hand. 
is  a  totality,  it  contains  everything  within  itself.  It 
therefore  has  no  position  and  is  not  two-sided. 

In  the  ordinary  study  of  Euclidean  surfaces  we 
frequently  make  use  of  certain  points  and  figures 
wdiich  are  not  on  these  surfaces  at  all.  We  talk  of 
normals,  of  tangent  lines  and  planes;  we  say  that 
surfaces  may  be  flexed,  shoved  about,  turned  over, 
or  displaced  in  any  direction  or  into  any  position. 
In  the  proof  of  equality  by  the  principle  of  congruent 
superposition,  it  is  i^ermitted  to  take  a  figure  fn^n  off 
the  plane  into  a  third  dimension,  obvert  it  and  bring 
it  back  again  into  congruetice  with  another  figure  in 
the  plane.     In  a  two-dimensional  space  none  of  the^e 


158  THE  NATURE  OF  SPACE 

things  are  possible ;  space,  as  we  have  said,  must  be 
complete  in  itself  and  independent  of  everything  else. 
But  the  self-dependence  of  any  space  can  only  be 
guaranteed  when  it  is  shown  that  measurements  can 
be  effected  and  a  complete  geometry  constructed 
without  leaving  it  at  all  or  even  appealing  to  any- 
thing external  to  it.  In  other  words,  it  must  be  pos- 
sible for  intelligent  beings  inhabiting  the  space  in 
question,  capable  of  moving  about  in  it  and  con- 
scious of  nothing  at  all  outside,  to  construct  a  geome- 
try true  for  any  and  every  part  of  it. 

As  we  have  seen,  a  Euclidean  plane  may  be 
wrapped  about  a  cylinder  or  a  cone,  it  may  be  trans- 
formed in  a  thousand  ways  so  far  as  its  relations  to 
a  third  dimension  are  concerned,  it  may  even  be 
v^^added  into  a  confused  mass  and  yet  so  long  as  no 
straining  or  distortion  of  its  internal  relations  occurs 
its  own  proper  geometry  remains  unchanged.  Hence 
considered  as  a  two-dimensional  space  it  is  the  same 
throughout.  For  intelligent  beings  whom  we  have 
supposed  to  inhabit  it,  who  cannot  leave  it  and  who 
know  nothing  external  to  it,  it  is  just  the  same  space ; 
its  metrical  properties  have  not  changed,  all  its  fig- 
ures remain  the  same,  and  its  straight  lines  are  al- 
ways and  everywhere  the  same  straight  lines. 

That  the  construction  of  such  a  geometry  is  possi- 
ble must  be  admitted  when  Riemann's  formula  for 
ds  comes  to  be  clearly  understood.  The  simple  con- 
ditions which  we  have  imposed,  that  the  curvature  of 


THE  NATURE  OE  SPACE  159 

the  space  in  question  shall  be  constant  and  that  its 
internal  relations  shall  remain  intact,  admit  the  ex- 
istence of  equal  spatial  quantities  in  different  places, 
and  this  is  all  that  Riemann's  arc  formula  requires. 
By  selecting  co-ordinates  which  shall  have  meaning 
in  the  space  with  whicii  we  are  dealing,  and  this 
can  always  be  done,  we  shall  have  all  the  conditions 
necessary  to  the  analytical  construction  of  the  re- 
quired geometry. 

The  same  truth  obviously  holds  for  non-Euclid- 
ean spaces  of  two  dimensions,  for  in  these  also  the 
necessary  conditions  of  free  mobility  and  of  un- 
changed internal  relations  are  fully  supplied.  Two- 
dimensional  manifolds,  both  Euclidean  and  non- 
Euclidean,  considered  individually  as  self-dependent 
totalities,  are  therefore  possible,  and  we  may  also 
have  many  varieties  of  these  provided  we  think  of 
them  as  unbounded  totalities  without  external  re- 
lations. We  need,  then,  simply  to  keep  in  mind  the 
fact  that  a  surface  in  a  space  of  any  variety  of  curva- 
ture is  quite  a  different  thing  from  a  two-dimen- 
sional space  of  that  same  variety.  The  surface 
always  requires  a  third  dimension;  the  two-dimen- 
sional space  does  not. 

In  the  case  of  tri-dimensional  spaces  the  same  prin- 
ciple holds  good.     Here,  too,  more  than  one  variety 
of  space  is  possible ;  but  in  this  case  the  problem  for 
imagination   is   an   impossible  one.     As   tri-dinicn 
sional  beings,  we  may  easily  represent  to  ourselves 


i6o  THE  MATURE  Of  SPACE 

ordinary  surfaces  as  two-dimensional  spaces  for  ra- 
tional beings  supposed  to  inhabit  them  and  to  be  con- 
scious only  of  their  internal  relations.  We  can  then 
see  from  the  standpoint  of  our  third  dimension  the 
numerous  modifications  which  can  be  made  to  take 
place  in  the  external  relations  of  these  surfaces  with- 
out in  any  sense  disturbing  their  internal  relations. 
Under  the  conditions  imposed  the  two-dimensional 
beings  inhabiting  these  spaces,  obviously,  could  not 
become  aware  of  any  of  these  external  changes ;  but 
supposing  them  to  be  endowed  with  intellects  like 
our  own,  some  Lobatchewsky  among  them  would 
sooner  or  later  work  out  two-dimensional  geometries 
which  would  not  hold  for  the  particular  space  inhab- 
ited by  him.  Three-dimensional  geometries  might 
also  be  constructed  by  this  same  two-dimensional 
genius.  And  yet  such  a  being  could  not  have  the 
necessary  sensory  experience  to  represent  to  himself 
what  these  spaces  really  are  as  judged  by  their  ex- 
ternal relations. 

As  we  stand  externally  related  to  these  two-dimen- 
sional beings,  so  it  is  conceivable  a  four-dimensional 
being  might  stand  related  to  us.  For  such  a  being 
various  modifications  in  the  external  relations  of  the 
space-world  known  to  us  and  of  the  objects  in  it 
might  occur  without  our  being  at  all  conscious  of 
the  change. 

We  cannot  in  any  a  priori  fashion  dogmatically 
denv  the  existence  of  a  four-dimensional  space-world 


THE  NATURE  OF  STAGE  16 1 

any  more  than  our  two-dimensional  beings  could 
deny  that  our  world  exists.  That  such  a  world  is 
inconceivable  in  the  sense  that  it  is  impossible  for  us 
to  imagine  it  must  be  admitted  on  the  ground  of 
actual  experience.  We  have  not  had  the  peculiar 
sense-experience  in  a  fourth  dimension  which  could 
render  any  sort  of  picture  of  it  possible.  We  can. 
of  course,  say  with  confidence  that  our  universe  as 
we  know  it  and  every  known  agency  in  it  is  confined 
by  some  law  of  its  being  or  at  least  of  our  knowing 
to  three  dimensions.  Our  space,  so  far  as  actual 
sense-experience  goes,  is  undoubtedly  tri-dimen- 
sional.  But  we  know  this  not  as  an  a  priori  neces- 
sity, as  Kant  contended,  but  as  an  empirical  fact. 
We  cannot,  therefore,  rightly  affirm  that  spaces  of 
higher  dimensions  than  ours  are  objectively  impos- 
sible or  even  inconceivable  in  the  sense  that  they  are 
not  rationally  construablcv^ 

Did  a  four-dimensional  world  exist  externally  re- 
lated to  our  own  as  ours,  in  turn,  stands  related  to 
the  two-dimensional  world  just  considered,  then,  as 
we  have  said,  it  would  be  possible  for  an  inhabitant 
of  that  world  to  witness  a  vast  number  of  external 
modifications  of  our  world  without  our  being  at  all 
conscious  of  any  change,  for  no  intrinsic  trans- 
formation would  actually  occur.  The  known  world 
would  remain  to  us  the  same  spatially  unchanging 
reality  that  it  now  appears  to  be. 

But  if  we  deny  the  actuality  of  a  fourth  dimen 


162  THE  NATURE  OF  SPACE 

sion,  the  modifications  which  we  have  supposed  to 
take  place  in  that  dimension,  of  course,  could  not 
occur.  In  that  event  would  more  than  one  variety 
of  tri-dimensional  space  be  possible?  This  question 
is  a  crucial  one.  Its  answer  determines  whether 
or  not  non-Euclidean  geometry  is  to  be  regarded  as 
having  any  peculiar  philosophical  significance.  The 
world,  as  our  actual  experience  reveals  it,  is  certainly 
tri-dimensional ;  judged  by  the  same  standard,  it  is 
also  Euclidean.  If,  then,  only  one  variety  of  tri- 
dimensional space  is  possible,  if  non-Euclidean  tri- 
dimensional geometry  really  demands  a  fourth  di- 
mension, the  so-called  non-Euclidean  spaces  are  in 
reality  not  spaces  at  all,  for  they  are  not  self-depend- 
ent totalities.  It  is  not,  then,  a  question  as  to 
whether  non-Euclidean  geometries  are  possible,  but 
a  question  as  to  whether  non-Euclidean  tri-dimen- 
sional spaces  are  possible.  It  is,  of  course,  possible 
to  construct  such  geometries  by  making  use  of  the 
idea  of  a  fourth  dimension,  just  as  we  ordinarily 
build  up  our  plane  geometry  by  frequently  referring 
to  figures  which  are  only  possible  in  a  third  dimen- 
sion ;  but  this,  of  course,  is  very  dififerent  from  estab- 
lishing the  possibility  of  non-Euclidean  tri-dimen- 
sional spaces. 

The  question,  then,  simply  reduces  to  this :  Are 
tri-dimensional  space-worlds  rationally  possible 
whose  internal  relations  considered  as  totalities  are 
essentially  different  from  each  other?    And  it  is  an- 


THE  NATURE  OF  SPACE  163 

swered  by  showing  that  the  geometries  of  such  spaces 
can  be  constructed  without  appeahng  to  a  fourth 
dimension.  This  can  be  done.  As  in  the  case  of 
two-dimensional  spaces,  we  have  here  also  all  the 
conditions  necessary  to  render  such  geometries  pos- 
sible. Indeed,  the  most  interesting  and  significant 
feature  of  non-Euclidean  solid  geometries  lies  in  the 
fact  that  they  are  just  as  independent  of  a  fourth 
dimension  as  is, Euclid  itself.  There  are,  to  be  sure, 
certain  facts  in  all  these  geometries  that  make  us 
wish  sometimes  for  a  fourth  dimension  and  the 
power  of  moving  into  it.  but  they  do  not  necessarily 
imply  this  dimension.  The  simple  principle  of  con- 
gruence fails,  for  example,  if  we  attempt  to  apply  it 
directly  in  proving  the  equality  of  two  Euclidean 
pyramids  wiiose  corresponding  parts  are  mutually 
equal  but  arranged  in  reverse  order.  As  we  have 
said,  the  analogous  theorem  in  plane  geometry  is 
proved  by  obverting  one  of  the  triangles  in  the  third 
dimension.  Were  there  a  fourth  dimension  and  had 
we  the  power  of  moving  into  it,  it  is  conceivable 
that  this  might  also  be  done  for  the  pyramids.  What 
would  happen  is  simply  this :  By  obverting  one  of 
the  pyramids  in  the  fourth  dimension  and  then  re- 
turning it  to  its  own  tri-dimensional  world,  its  rela- 
tions to  the  other  objects  af  this  world  are  changed 
in  a  way  that  is  wholly  impossible  so  long  as  we 
confine  it  to  three  dimensions.  But  the  internal  rela- 
tions of  the  pyramid  itself,  as  in  the  observed  case  of 


i64  THE  NATURE  OF  SPACE 

the  triangle,  remain  entirely  unaltered.  The  self- 
identity  of  the  figure  is  retained.  But  as  we  have 
said,  these  facts  cannot  be  regarded  as  implying  the 
logical  dependence  of  Euclid,  or  of  non-Euclid,  upon 
a  fourth  dimension. 

Granting,  then,  as  we  must,  the  possibility  of  in- 
ternally different  tri-dimensional  spaces,  we  need  to 
inquire  briefly  into  their  mutual  relations  and  what 
they  imply. 

Their  differences,  as  already  indicated,  are  chiefly 
metrical.  Hence  in  all  of  them  those  qualitative 
characteristics  which  are  presupposed  in  measure- 
ment are  implied.  It  is  these  characteristics  which 
render  possible  in  each  individual  case  those  peculiar 
internal  relations  which  constitute  the  essential  na- 
ture of  space  as  a  totality.  Whatever  intrinsic  qual- 
ities are  necessary  to  all  measurement,  as  such,  these 
different  spaces  must  possess  in  common.  Just 
what  their  common  elements  are  we  shall  postpone 
for  the  moment  and  consider  here  their  differences 
merely.  These  differences  are  bound  up  in  the 
modifications  of  meaning  which  are  possible  in  the 
peculiar  conception  of  curvature  to  which  we  have 
often  referred. 

As  already  pointed  out,^  curvature  as  applied  to 
space  represents  an  intrinsic  property  of  the  particu- 
lar space  in  question  and  does  not  necessarily  imply 


Chap.  II.,  p.  33' -3^        ff-^  I'j 


I  THE  NATLRt  Ul-  SPACE  165 

a  higher  dimension.  Measurement  ni  the  exact 
sense  required  by  geometry  is  not  possible  unless  the 
space  in  which  it  occurs  possesses  the  property  of 
perfect  homogeneity.  It  must  permit  the  free  mo- 
biHty  of  a  rigid  standard,  and  its  metrical  peculiari- 
ties, whatever  they  are,  must  be  true  of  it  as  a  whole. 
This  is  what  is  meant  by  saying  that  its  curvature  is 
constant. 

If.  then,  geometrical  spaces  of  the  same  number 
of  dimensions  may  be  of  different  varieties,  it  follows 
that  their  space-constants  must  also  be  different,  and 
it  is  this  difference  that  we  need  to  examine.  It  is 
almost  universally  regarded  as  a  quantitative  one. 
Men  talk  as  though  space  constants  might  be  ar- 
ranged as  a  series  of  quantities  whose  ratios  could 
be  exactly  determined  by  the  use  of  a  common  stand- 
ard of  measurement.  This  is  wrong.  The  law  of 
exact  homogeneity  forbids  it.  Strictly  speaking, 
quantitative  differences  can  only  exist  in  the  same 
space,  for  here  alone  the  exact  measurement  of  all 
figures  by  the  same  rigid  standard  is  possible.  Dif- 
ferent spaces  of  the  same  number  of  dimensions  do 
not  admit  of  the  free  mobility  throughout  their  whole 
extent  of  the  same  rigid  body ;  if  they  did  they 
could  not  be  metrically  distinguished.  But.  as  we 
have  said,  it  is  in  metrical  properties  that  Euclidean 
and  non-Euclidean  space-conceptions  are  different ; 
projectively  treated  these  differences  disa|)pear. 
Space-constants  must  therefore  be  regardefl  as  stand- 


i66  THE  NATURE  OF  SPACE 

ing  related  to  each  other  in  a  manner  similar  to  what 
obtains  in  different  shades  of  the  same  color.  They 
are  qualitatively  different.  They  have  much  in  com- 
mon, but  are  at  the  same  time  so  unlike  as  to  make 
exact  geometrical  measurement  impossible.  There 
exists  no  common  standard ;  hence,  strictly  speaking, 
we  cannot  say  that  any  space-constant  is  greater  or 
less  than  another.  Logical  difficulties  of  a  similar 
nature  obviously  follow  the  attempt  to  regard  the 
curvature  of  any  geometrical  space  as  a  variable.^ 
This  involves  the  assumption  of  an  absolute  position 
and  consequently  denies  the  relativity  and  homogene- 
ity which  strict  geometrical  measurement  always 
requires.  Such  a  variable  curvature  is  represented 
for  example  by  the  surface  of  an  tgg,  in  which,  gen- 
erally speaking,  it  is  not  possible  to  make  one  piece 
fit  upon  another  without  distortion.  Regarded  as  a 
space,  it  is  not  homogeneous. 

Here  again  we  are  forced  to  observe  the  necessary 
disparity  between  the  facts  of  experience  and  the  ab- 

2  It  is  not  necessary  to  enter  into  an  extended  argument  in 
support  of  this  position.  Tlie  curvature  of  space  might  be 
regarded  as  variable  if  the  law  of  its  change  were  known. 
This  is  practically  what  is  involved  in  Cayley's  idea  of  con- 
structing non-Euclidean  geometries  in  Euclidean  spaces  by 
simply  modifying  the  notion  of  distance.  It  is  also  suggested 
by  Erdmann  (^Die  Axiome  der  Geometric,  Leipzig,  1877). 
But  while  such  a  law  of  change  may  be  assumed  we  are  left 
absolutely  without  any  means  of  determining  what  it  is  or  even 
of  detecting  its  presence.  Diflferences  of  magnitude  where 
comparison  can  not  reveal  them  are  manifestly  at  variance 
with  the  very  notion  of  magnitude. 


THE  NATURE  OF  SPACE  167 

solute  exactitude  demanded  by  geometry.  Geomet- 
rical measurement  is  of  course  impossible  without 
the  assumption  of  absolute  homogeneity;  but  prac- 
tical measurement,  the  only  kind  possible  in  the  world 
as  we  know  it,  can  be  performed.  It  is  also  clear,  in 
support  of  our  general  position,  that  the  question 
whether  the  actual  space-parameter  of  our  world, 
when  assumed  to  be  constant,  is  Euclidean  or  non- 
Euclidean,  cannot  be  settled  by  any  mere  appeal  to 
a  rigorous  geometry.  For,  as  we  have  just  seen, 
the  metrical  comparison  of  different  space-parame- 
ters upon  such  a  basis  cannot  occur.  We  cannot 
travel,  so  to  speak,  with  our  metrical  standard  from 
one  space  to  another. 

We  must  remember,  however,  that  spatial  homo- 
geneity is  merely  a  conception ;  as  to  its  nature  and 
origin  it  does  not  differ  from  other  conceptions.  It- 
is  simply  an  idealization  of  certain  familiar  facts  of 
our  spatial  experience.  It  is  the  facts  that  determine 
the  meaning  of  homogeneity,  and  not  homogeneity 
that  determines  the  facts.  Geometry  must  fit  experi- 
ence, not  experience  geometry. 

To  reverse  this  order  is  a  crime  against  knowledge 
which  has  been  too  often  committed.  The  homo- 
geneity of  space  is  in  fact  a  complex  idea  and  admits 
of  being  variously  interpreted.  This  is  shown  by 
the  existence  of  different  metrical  geometries  in 
which  homogeneity  is  so  conceived  as  not  to  admit 
of  the  free  mol)ility  of  the  same  rigid  body  in  the 


i68  THE  NATURE  OF  SPACE 

different  cases.  It  is  not,  then,  a  question  as  to 
which  of  these  conceptions  is  to  be  accepted  as  the 
absolute  truth,  but  as  to  which  one  accords,  and  will 
continue  to  accord,  best  with  man's  actual  experi- 
ence. Experience  itself  must  decide  this  matter,  and 
even  its  decision  must  always  be  regarded  as  only 
approximate.  Homogeneity,  then,  whatever  par- 
ticular view  we  take  of  it,  is  a  different  conception 
from  mere  logical  anyness  with  which  it  is  often  con- 
fused. It  bears,  in  some  sense,  the  stamp  of  actual 
experience  as  is  seen  in  the  fact  that  the  geometrical 
figures  which  it  admits  appear  to  have  a  distinct 
character  and  reality  of  their  own.  If  one  does  not 
believe  this  let  him  take  any  material  body  and  en- 
deavor to  cut  from  it  a  Euclidean  regular  poly- 
hedron of  seven  plane  faces  and  his  doubts  will 
vanish.  But  why  is  this  figure  impossible?  It  is 
certainly  not  because  of  any  obstacles  which  the  ma- 
terial itself  can  offer,  for  this  may  be  sawed  through 
in  one  direction  as  easily  as  another.  The  truth  is 
the  lines,  angles,  and  planes  enter  into  the  structure 
of  such  geometrical  figures  with  all  the  meaning 
which  our  actual  tri-dimensional  experience  has 
given  them.  Indeed,  it  ought  now  to  be  perfectly 
clear  that  lines  and  planes  as  ordinarily  conceived 
are  not,  in  reality,  one  and  two-dimensional  objects 
respectively.  They  are  this  and  more.  To  give 
them  their  full  meaning  they  require  a  spatial  experi- 
ence of  a  tri-dimensional  order.     The  straight' line 


THE  NATL' HE  UF  SPACE  Kxj 

as  it  exists  in  the  average  man's  consciousness  is  per- 
haps best  defined  as  a  Hne  which  looks  the  same  from 
every  point  not  in  it,  but  this  definition  clearly  impli- 
cates three  dimensions.  If,  however,  we  define  it  as 
the  shortest  distance  between  two  points,  and  it  is 
then  pointed  out  that  arcs  of  great  circles  on  a  sphere 
meet  this  requirement,  we  are  not  satisfied;  for  this 
is  evidently  not  what  we  mean.  It  is  the  chord,  we 
say,  not  the  arc,  that  stands  as  a  true  representative 
of  our  idea  of  the  straight  line.  Thus  again  we 
must  enter  the  third  dimension  before  we  can  decide 
the  matter.  If  an  ordinary  plane  be  conceived  as 
wrapped  about  a  cone,  it  remains  the  same  two- 
dimensional  object  that  it  was  before,  but  it  is  for 
us  no  longer  a  plane.  To  be  a  plane  it  must  sustain 
certain  definite  relations  to  a  third  dimension. 

It  is  therefore  obvious  that  the  peculiar  form  of 
our  tri-dimensional  experience  has  given  us  these 
conceptions  of  the  straight  line  and  the  plane.  The 
moment  we  begin  to  deal  with  definite  geometrical 
figures  of  any  sort  we  are  considering  combinations 
of  relations  whose  peculiar  character  as  combinations 
is  determined  by  certain  conceptions  which  are  thus 
grounded  in  experience.  They  are  certainly  not  ra- 
tional necessities.  Homogeneity  of  some  sort  is  an 
essential  characteristic  of  any  space-conception  which 
will  admit  of  the  exact  quantitative  determinations 
of  metrical  geometry.  The  particular  combinations 
of  this  conception  and  those  assumptions  which  are 


170  THE  NATURE  OF  SPACE 

needed  in  each  case  to  define  accurately  and  com- 
pletely the  peculiar  foundations  of  the  different 
geometries  must,  of  course,  be  regarded  as  necessary 
to  the  particular  space-conceptions  upon  which  these 
geometries  are  founded;  but  necessities  of  thought, 
presupposed  in  the  very  possibility  of  any  spatial  ex- 
perience whatever  or  even  of  experience  as  we  actu- 
ally have  it,  they  certainly  are  not. 

Underneath  these  conceptions  and  implicit  in  them 
the  true  category  of  space  is  to  be  found.  As  a 
"  differentiating  principle  "  ^  of  some  sort,  both  sub- 
jectively and  transsubjectively  operative,  externaliz- 
ing and  at  the  same  time  uniting  a  world  of  co-exist- 
ing selves  and  things,  this  category  is  necessary  to  all 
human  cognition  or  mental  representation  of  the 
world  of  reality.  As  such  we  cannot  escape  it,  for 
we  can  think  of  no  entity  to  which  it  does  not  apply. 
It  was  the  unavoidable  presence  of  space  in  this 
meaning  of  the  word  that  seems  to  contradict  flatly 
much  that  was  said  at  the  beginning  of  this  chapter, 
where  we  represented  geometrical  spaces  as  totalities 
and  yet  spoke  of  them  as  though  they  stood  related 
somehow  to  realities  to  which  as  spaces  they  could 
not  apply.  The  confusion  at  that  stage  of  the  dis- 
cussion seemed  necessary.  We  could  not  avoid  it. 
It  is  now  possible  to  make  clear  what  was  meant. 
The  space-conceptions  of  geometry  are  not  catego- 

3  See   Professor  Ladd's  admirable  chapter  on   "  Space  and 
Motion,"  in  A  Theory  of  Reality,  New  York,  1899,  pp.  214-252. 


THE  NATURE  OF  SPACE 


171 


ries,  not  necessities  of  thought  or  reahty ;  they  imply 
the  space  category,  it  is  true,  but  they  also  imply 
more.  Stamped  upon  them  are  certain  marks  of  an 
empirical  origin  which  are  grounded  in  the  peculiar 
nature  of  our  human  spatial  experience.  To  deter- 
mine these  marks  which,  so  far  as  we  can  see,  might 
have  been  different ;  to  separate  them  out  and  if  pos- 
sible to  get  directly  at  the  peculiar  nature  of  the  cate- 
gory itself,  was  the  difficult  problem  which  we  there 
had  in  mind.  When  we  limit  the  number  of  dimen- 
sions and  introduce  particular  figures  and  special 
metrical  considerations  we  no  longer  deal  with  the 
"  pure  "  category  of  space,  as  it  must  lie  implicit  in 
any  rational  being. 

We  conclude,  then,  that  the  actual  characters  of 
our  space  experience  so  far  as  they  involve  tri-dimen- 
sionality,  the  parallel  postulate,  &c.,'are  empirical  in 
the  same  sense  in  which  any  notable  feature  of  our 
present  human  experience  is  empirical.  As  it  is  an 
empirical  fact  that  there  are  many  different  men 
with  different  minds  or  that  we  can  grasp  in  one  act 
of  attention  only  a  narrow  range  of  facts,  so  it  is 
an  empirical  fact  that  the  above-mentioned  charac- 
ters belong  to  our  space-consciousness.  Other  be- 
ings may  have  other  space  experiences.  We  our- 
selves may  some  day  acquire  other  such  experiences. 
There  is  no  demonstrable  necessity  accessible 
regarding  the  matter,  so  far  as  the  details  are  con- 
cerned. 


172  THE  NATURE  OF  SPACE 

The  only  a  priori  manifold  at  present  definable  in 
Kant's  sense  of  a  priori  seems  to  be  a  manifold  con- 
stituted by  a  totality  of  logical  classes  or  distinctions 
of  any  similar  sort.  The  constitution  of  such  a  com- 
plete system  of  logical  entities  must  be  implicitly 
known  to  any  rational  being.  It  depends  upon  the 
fundamental  illative  relation  expressed  by  such  state- 
ments as  "  a  implies  b,"  or  that  "  a  is  subsumed  un- 
der b."  A  world  in  which  this  relation  has  full  sway 
contains  the  "  negation  "  of  any  of  its  own  terms. 
It  contains  series  of  subsumptions  and  also  certain 
combinations  whereby  its  terms  are  grouped  into 
wholes.  Such  a  system  has  a  formal  structure 
which  Kempe  ^  has  shown  includes  all  the  types  of 
order  which  appear  in  every  sort  of  geometry  of  n 
dimensions.  Any  space  you  please  may  be  viewed 
as  a  selection  from  amongst  the  entities  of  such  a 
system  made  by  adding  certain  arbitrary  postulates 
to  these  fundamental  logical  ones. 

The  connection  between  this  a  priori  logical  mani- 
fold and  the  empirical  space  of  our  own  experience 
lies  in  the  fact  that  the  space-aspect  of  experience 
is  the  one  which  jnost  definitely  implies  and  is  im- 
plied by  our  power  to  co-ordinate  our  activities  so 
that  "  a  leads  to  b  leads  to  c,"  &c.  It  is  that  aspect 
which    enables    us    to    introduce    illative    relations 


■*  "  Relation  Between  the  Logical  Theory  of  Classes  and  the 
Geometrical  Theory  of  Points."  Proceedings  of  the  London 
Mathematical  Society  for  1890,  Vol.  21. 


THE  NATURE  OF  SPACE  173 

among  acts  and  systems  of  acts  of  our  own  (acts 
actual  and  acts  possible). 

That  this  aspect  of  experience  exists  is  an  em- 
pirical fact.  What  correlations  of  acts  it  permits 
and  how  it  permits  them  are  also  empirical.  All  the 
details  are  empirical.  But  if  it  is  to  permit  such  a 
system  at  all,  it  has  to  conform  to  the  general  type 
of  the  illative  relation  and  its  parts  viewed  as  co- 
existent must  be  related  to  each  other  in  accordance 
with  the  general  type  of  an  illative  relation.  "  No 
actualization  of  the  space-principle  is.  therefore,  pos- 
sible either  from  the  point  of  view  of  its  subjective 
origin  or  of  its  trans-subjective  applicability,  unless 
this  principle  itself  is  conceived  of  as  the  mode  of 
the  action  of  one  all-differentiating  and  yet  all  unify- 
ing Force,"  •'  functioning  in  a  way  that  is  essentially 
law-abiding  and  rational. 

3  Ladd's  A  Theory  of  Reality,  p.  252. 


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